# Limits and convergence.

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Chapter 2 Limits and convergence. 2.1 Limit points of a set of real numbers Limit points of a set. DEFINITION: A point x R is a limit point of a set E R if for all ε > 0 the set (x ε,x + ε) E is infinite. ex Prove that x is a limit point of E if ε > 0, y E,0 < x y < ε; that is: for all ε > 0 the set (x ε,x + ε) contains a point y E, y x Limit points of a sequence DEFINITION: A sequence of real numbers, aka a numerical sequence, is a map n a n R defined for n N, n Z or, more generally, n A Z. The notation is {a n } n A. A sequence is finite if the index set A is finite, otherwise it is infinite. Note that the sequence is finite or infinite depending on the index set and not on the size of the range. Thus {a n } n N is an infinite sequence even if a n = 1 for all n N. DEFINITION: A point x R is a limit point of a numerical sequence {a n } n=1, if for all ε > 0 the set {n : a n x ε} is infinite. Thus in the example above, where a n = 1 for all n, the point x = 1 is a limit point of the sequence {a n } but is not a limit point of the set {1} which is the range of the sequence Convergence of sequences (of real numbers): Convergent sequences; subsequences; 10

2 2.2. Completeness, Bolzano Weierstrass Completeness, Bolzano Weierstrass The least upper bound of a set DEFINITION: The number b is the supremum aka the least upper bound of a set E R, written b = supe, if it is an upper bound for E and no number smaller that b is an upper bound for E: formally a. x E,x b, and b. if c R, c < b, then y E, y > c, (i.e., c is not an upper bound for E). Least upper bound property; consequences; Exercises: quantifiers Move to the appendix: countability of Q vs. uncountability of R Completeness. Cauchy sequences and completeness; Cauchy criterion. ====== Theorem. the completeness axiom for R as stated sometimes, namely: a. Every nonempty subset S of R that is bounded above has a least upper bound in R, (denoted sups). is equivalent to the statement b. Every Cauchy sequence of real numbers has a limit in R. PROOF: We need to show that each statement implies the other a. = b. Assume the completeness axiom. Preparation: Observe that for a bounded monotone non-decreasing sequence y n one has limy n = sup{y n } (The sup exists by the Completeness axiom. By the definition of sup there exist, for any ε > 0, values of N such that y N > sup{y n } ε and if n > N we have y N y n sup{y n } ). Similarly for a bounded monotone non-increasing sequence y n one has limy n = inf{y n }. The completeness axiom now implies that every

3 12 2. Limits and convergence. bounded monotone sequence y n of real numbers has a (real number) limit. This implies the existence, for any bounded sequence {x n }, of the limits limsupx n = lim m sup n>m x n and liminfx n = lim m inf n>m x n. (since y m = sup n>m x n and z m = inf n>m x n are monotone and bounded.) Let {x n } R be a Cauchy sequence; it is clearly bounded. Write S = limsupx n, s = liminfx n then S s and we claim that in fact S = s so that limx n = S and {x n } has a limit in R. We show that S = s by showing that the opposite assumption in not consistent with the Cauchy condition on {x n }. Assume S s > 0 and set ε = S s 3 ; there are arbitrarily large values n, m such that x n < s + ε and x m > S ε so that x m x n > ε, which contradicts the assumption that {x n } is a Cauchy sequence. b. = a. Assume Every Cauchy sequence of real numbers has a limit in R. Let A be a set which is bounded above, and let x 1 be an upper bound. If x 1 = supa we are done. Otherwise let k be the smallest integer such that x 1 1 k is an upper bound (for A) and set x 2 = x 1 1 k. Repeat, defining x n+1 = x n kn 1 where k n is the smallest integer such that x n 1 k n is an upper bound for A; (if at any stage x n = supa we are done ). The sequence {x n } is bounded below (by A) so that k 1 n < Theorem (Bolzano Weierstrass). A bounded infinite set of real numbers has limit points. PROOF: Let E be an infinite set contained in the (finite) interval I = [a,b] R. At least one of the intervals I 0 =[a, a+b 2 ] and I1 =[ a+b 2,b] contains infinitely many elements of E. Write J 1 = I 1 if I 1 E is infinite; otherwise set J 1 = I 0. J 1 E is infinite, and denoting by c 1 is the right end point of J 1, the half line (c,+ ), has finite intersection with E. Let J 2 be the right hand half of J 1 if its intersection with E, J 2 E, is infinite; otherwise set J 2 equal to the left hand half of J 1. The choice guarantees that J 2 E is infinite, while E has only finitely many points

4 2.2. Completeness, Bolzano Weierstrass 13 to the right of J 2. We continue in the sam manner: assuming that we have J m defined so that J m is half of J m 1, J m E is infinite, and E has only finitely many points to the right of J m, we split J m into two equal intervals and set J m+1 equal to the right half of J m if the right half has infinitely many points of E, and equal to the left half of J m otherwise. We denote by c m the right end point of J m and observe that {c m } is a Cauchy sequence since if m j > M, j = 1,2, both c m2 and c m1 are contained in J M so that c m2 c m1 < I 2 M. By the Cauchy criterion, {c m } is convergent. Write c = lim m c m, and verify that [c + ε, ) E is finite for every ε > 0, while [c ε, ) E is infinite. Thus, c is a limit point of E, and in fact the biggest limit point, i.e. c = limsupe Open covers, Heine-Borel. DEFINITIONS: An open cover of a set E in a metric space (X,ρ) is a collection of open sets {O α } α A, such that E α A O α. A subcover of an open cover {O α } α A (of E), is a subcollection, i.e., {O α } α B where B is a subset of the index set A and such that E α B O α. The subcover is a finite subcover if the corresponding index set B is finite. The set E is compact if every open cover of E has a finite subcover. Theorem (Heine-Borel). a. A set E R is compact if and only if it is both bounded and closed. b. A set E R is compact if and only if every infinite subset thereof has limit points in E. PROOF: a. Assume E compact. Let O n =( n,n), then O n covers R, hence E. A finite subcover has a biggest element and it is bounded, so E is bounded. We prove that E is closed by showing that its complement is open. If y E the sets O n = {x : x y > 1 n } are open and their union covers everything but y, hence covers E. If the union of a finite subcollection n N O n = O N covers E, then the neighborhood {x : x y < 1 N } of y is disjoint from E, so that the complement R \ E is open.

5 14 2. Limits and convergence. We prove the converse by contradiction. Let E be bounded and closed, and let A = {O α } α A be an open cover of E. Let I =[a,b] be a finite interval containing E. Assume that there is no finite subcover of E for any finite subset B A the family {O α } α b fails to cover E. As in the proof of the Bolzano-Weiestrass theorem we split I into two equal intervals, I 0 and I 1 and observe that if both I 0 E and I 1 E can be covered by a finite subfamily of A, then the union of the two finite families is a finite cover for E. It follows that at least one of I 0, I 1 has the property that its intersection with E admits no finite subcover. Denote by J 1 either of I 0, I 1 that has this property. By the same argument we have an interval J 2 J 1, which is either the left half or the right half of J 1 that has the property that J 2 E does not admit a finite subcover from A. Continuing, we obtain a sequence {J n } of intervals, each J n is either the left or the right half of J n 1, and for every n the set J n E does not admit a finite subcover from A and in particular, is not empty. Let x n J n E. The sequence {x n } is a Cauchy sequence hence convergent, and since E is closed, x = lim n x n E. Since x E, there exists α A such that x O α, and since O α is open it contains an open interval centered at x and therefore contains J n if n is big enough, contradicting the assumption that J n E is not contained in a finite union of O α s. b. A set E R such that every infinite subset thereof has limit points in E is bounded, otherwise it contains a sequence {x n } such that x n > n that clearly has no limit points; and closed, otherwise it contains a sequence {x n } that converges to point in the complement of E, hence has no limit points in E Total boundedness. DEFINITION: A metric space (X, ρ) is totally bounded if for every ε > 0 there exists a finite set {x j } n j=1 X such that B(x j,ε)=x. ex Prove that a compact metric space is totally bounded and complete. ex Prove that a totally bounded complete metric space is compact.

6 2.3. Series 15 ex Show that the material of this subsection extends verbatim to subsets of R d for any d N. 2.3 Series Series: basic convergence. An infinite series is an expression of the form S = j J a j where J is an ordered infinite index set. We shall focus on the most common case 1 J = N, in which case we write S = j=0 a j. The partial sums of the series are the expressions S N = N j=0 a j. The partial sums of a series are real numbers computed as usual by addition. The sum of the series S can not be computed by (repeated) addition alone, and is commonly defined as the limit, for N, of the partial sums. (2.3.1) S = a j = lim N j=0 N j=0 This definition is consistent with what we expect from a series that has only finitely non-zero elements. DEFINITION: The series S = j=0 a j is convergent if the sequence {S N } of partial sums converges, and the sum of the series is given by (2.3.1). EXAMPLE: Geometric series. These are series of the form 0 an where a < 1. Since S N = N 0 an = 1 an+1 1 a the series converges and its 1 a sum S = lim N+1 N 1 a = 1 1 a Cauchy criterion of convergence. Since the convergence of a series S is defined by means of the convergence of the sequence of its partial sums, conditions that guarantee the latter automatically guarantee the former. For example: Proposition. A necessary and sufficient condition for the convergence of S is that {S N } be a Cauchy sequence. a j. 1 Another common case is J = Z, in which case we write j= a j.

7 16 2. Limits and convergence. By definition, the condition that {S N } be a Cauchy sequence is: for every ε > 0 there exists N = N(ε) such that if m > l > N then S m S l = m l+1 a n < ε. In particular, if a n converges then a n Series with nonnegative terms. When all the entries a n of a series S are nonnegative, the partial sums S N = N 0 a n form a monotone nondecreasing sequence, and hence converge to a (finite) limit if and only if the sequence is bounded. Observe that, being monotone, the sequence is bounded if it has a bounded infinite subsequence Conditional and absolute convergence. DEFINITION: A series S = 0 a n converges absolutely if the series of the absolute values of its summands, 0 a n converges. A convergent series that does not converge absolutely is said to converge conditionally. Proposition. If the series S = 0 a n converges absolutely then it converges. The proof is an immediate consequence of the Cauchy criterion, since (2.3.2) m a n l+1 m l+1 a n. The same observation gives the comparison test: Proposition (Comparison test). Assume b n a n. If a n converges then so does b n. The comparison test can be used in a slightly more general context. Consider the series S = 1 n a, a > 1. The terms are nonnegative and, in order to prove convergence, it suffices to prove the partial sums bounded. This is seen by considering the blocks B m = 2m+1 2 m +1 n a. B m is the sum of 2 m terms, each of which is bounded by 2 ma so that B m 2 (1 a)m, and comparing to the geometric series with ratio 2 1 a < 1 we obtain B m <. Since the partial sums of B m are partial sums of S, the latter are bounded and S converges.

8 2.3. Series 17 ex Prove that S = 1 n a, a 1 does not converge. ex Prove that S = 1 n 1 (logn) a converges if and only if a > 1. Hint: For 2 m < n 2 m+1 we have n 1 (logn) a = O(2 m m a ) Rearrangements. The sum of a finite number of real numbers does not depend on the order in which they are added this is the commutative law. For infinite series, which use addition but also use a limiting process, the order may play a big role. DEFINITION: A permutation of N is a 1 1 map of N onto itself. A rearrangement of a series n=0 a n is a series of the form n=0 a σ(n), where σ is a permutation of N. Theorem. a. If the series a n is absolutely convergent, then every rearangement thereof is absolutely convergent, and to the same sum. b. If the series a n converges conditionally then, given a number c R, there exist rearrangements thereof converging to c. PROOF: a. Denote the sum of the series by a. Let ε > 0 and let N = N(ε) be such that N a n < ε. Observe that if J N, and J [0,N] then a n J a n < ε. If σ is a permutation of N then, given N, there is an M(N) such that (for all M > M(N)) {σ(n) : n < M} [0,N] and we have a M j=0 a σ( j) < ε. b. The conditional convergence guarantees that lima n = 0 and a n =. Split the sequence {a n } into two: the subsequence I + of nonnegative elements and the subsequence I of negative elements. We have (2.3.3) n I + a n =+ and n I a n =. We create a permutation σ by choosing successively elements, in the order they appear, from either I + or I. Assume that c 0. We start by taking (the first) elements from I + stopping when the sum of the chosen elements exceeds c; this is turning point 1. We now take elements from I until the sum goes below c. This is turning point 2. We take the next elements from I + until the (cumulative) sum surpasses c (turning point 3), and then form I until the sum is below c, and by (2.3.3) this procedure can be repeated indefinitely.

9 18 2. Limits and convergence. The partial sums that correspond to turning points are obtained by adding some positive a n to a value smaller that c or by adding some negative a n to a value bigger that c; all other partial sums are between two that correspond to successive turning points. Given ε > 0, once all the terms a n of absolute value exceeding ε are used up, the partial sums of the rearranged series are within ε from c. ex Let a n be conditionally convergent. Prove that the set of limit points of the partial sums of a σ(n), for any permutation of N, is a closed, not necessarily bounded, interval (including a single point). Moreover, any closed interval, as above, is the set of limit points of the partial sums of a σ(n) for some permutations σ of N Summability. It is often very useful to consider infinite series whose partial sums fail to converge. The following method to assign sum to (some) infinite series that fail to converge is the Cesàro (C,1) summability. It extend the standard definition of sum as the limit of the partial sums of the given series to one that assigns a sum to a wider class of series, and does it in a consistent way it gives the same value to convergent series as the standard definition. DEFINITION: A means A series a n is (Cesàro) (C,1)-summable to the sum S 0 + S S n (2.3.4) lim = A n n + 1 Theorem. Let S = a n be a convergent series, S = lim n S n, then S is (C,1)-summable to the same value S. PROOF: The entries a n and the partial sums are bounded. Write σ n = 1 n+1 n j=0 S j. Given ε > 0, there exists N such that for n > N liminfs j ε S n limsups j + ε. It follows that for all m N m infs j + m N m liminfs j ε 1 m m j=1 S j N m sups j + m N m limsups j +ε.

10 2.4. Continuity (R-valued functions ) 19 As m, N m N m 0 and m 1, so that liminfs n ε liminfσ n limsupσ n limsups n + ε. Since ε is arbitrary we obtain (2.3.5) liminfs n liminfσ n limsupσ n limsups n Since we assume the series to converge, all the inequalities above are in fact equalities and limσ n = lims n On the other hand if a n =( 1) n we have S n = 1 if n is even, and S n = 0 for n odd. S n clearly does not have a limit and yet, σ n 1/ Continuity (R-valued functions ) We consider real-valued functions defined on a set E in a metric space (X,ρ). There is no loss of the picture in assuming that (X,ρ) is R with ρ(x,y)= x y, or R d with the Euclidean metric. The metric on E is the metric induced by (X,ρ) Limits of a function at a point. Assume that x is a limit point of a set E in a metric space (X,ρ), and f a real-valued function defined on E. DEFINITION: limsup y x, y E f (y)=lim ε 0 sup{ f (y) : y B(x,ε) E} liminf y x, y E f (y)=lim ε 0 inf{ f (y) : y B(x,ε) E}. Observe that the limits exist since sup{ f (y) : y B(x, ε) E} and inf{ f (y) : y B(x,ε) E} are monotone in ε. The reference to the set E is omitted if E is understood The oscillation o f (x) of a function f at a point x. The oscillation is defined by (2.4.1) o f (x)=limsup y x An equivalent definition is given by f (y) liminf y x f (y) (2.4.2) o f (x)= limsup f (y 1 ) f (y 2 ) y j x, j=1,2

11 20 2. Limits and convergence. in other words, o f (x) is the least upper bound of f (y 1 ) f (y 2 ) with y j arbitrarily close to x. That means that o f (x) > a if for all ε > 0 there exist y 1 and y 2 in B(x,ε) such that f (y 1 ) f (y 2 ) > a. Remark: The definition of o f in terms of limsup and liminf, (2.4.1), uses the order on R. The second definition uses the metric rather than the order and while it is equivalent for real-valued functions, it applies equally, with f (y 1 ) f (y 2 ) replaced by ρ( f (y 1 ), f (y 2 )) to mappings into a general metric space. ex Prove that for any real-valued f and any constant B, the set {c : o c ( f ) B} is closed Continuity at a point. DEFINITION: A real-valued function f defined on a set E is continuous at a point x 0 E if for all ε > 0 there exists δ > 0 such that if y E and y x 0 < δ then f (y) f (x 0 ) < ε. An equivalent definition is: f is continuous at x 0 if o f (x 0 )=0. The number δ depends on the complete configuration: the set E, the function f, the point x 0 and the number ε. We can be explicit by writing δ = δ(e, f,x 0,ε). ex Assume that both f and g are functions defined on a set E and both are continuous at a point x 0 E. Prove that f + g and fg are continuous at x Continuity on a set. DEFINITION: A real-valued function f defined on a set E is continuous on E if it is continuous at every x E. We denote by C R (X) the set of all 2 real-valued functions defined and continuous on X. ex Prove that if f, g C R (X) then f + g and fgare in C R (X). 2 The symbol C(X) commonly denotes the space of all continuous complex valued functions on X.

12 2.4. Continuity (R-valued functions ) Uniform continuity. DEFINITION: A real-valued function f defined on a set E is uniformly continuous on E if for all ε > 0 there exists δ = δ(e, f,ε) > 0 such that if if y,x E, y x < δ then f (y) f (x) < ε. Uniformity means that for every ε > 0 one has a positive δ common to all the points x in E. Theorem. Let f be a real-valued function defined on a compact set E R, and continuous at every x E. Then a. f is bounded on E. b. f is uniformly continuous on E. c. The range of f is closed; in particular f assumes the values sup x E f (x) and inf x E f (x). The continuity of f implies that for every ε > 0, and every x E, there exists δ = δ(x) > 0 such that if y E and y x < 2δ, then f (y) f (x) < ε/2. PROOF: a. The open balls B(x,δ(x)), x E form an open cover of E and since E is compact, there is a finite subcover, that is, a finite set G E such that x G B(x,δ(x)) = E. The boundedness of f follows from the fact that for all y E, f (y) ε + max x G f (x). b. To prove the uniform continuity we have to show that for every ε > 0 there exists δ > 0 such that if y,z E and z y < δ, then f (y) f (z) < ε. Keeping the notation we have above, we claim that δ = min x G δ(x) has the needed property. Let y,z E, z y < δ. Let x G be such that y B(x,δ), and hence z B(x,2δ). Now, f (y) f (x) < ε/2 and f (z) f (x) < ε/2 so that f (y) f (z) < ε. c. Let y be a limit point of the range of f. Let x n E be such that lim n f (x n )=y and let x 0 be a limit point of {x n }, (its existance guaranteed by the Bolzano-Weierstrass theorem) then f (x 0 )=lim n f (x n )=y, and y is in the range of f.

13 22 2. Limits and convergence. The compactness assumption is crucial as can be seen by the following examples: E =(0,1), f (x)=1/x, (E is not closed). E = R, f (x)=x 2, (E is not bounded). ex Prove that the range of a continuous function on a compact set is closed. (Since, by a. above, the range is also bounded this shows that it is in fact compact.) Hint: Repeat the proof of c. above. ex Let f be a continuous map from a compact metric space (X,ρ) into a metric space (Y,ρ ). Assume that f is a bijection (1-1 map) form its domain X onto its range Y = f (X) Y. Prove that the inverse map f 1 is continuous on Y. ex Prove that a real-valued function whose domain is a metric space (X,ρ) is continuous on X if and only if the pre-image f 1 (O)= {x X : f (x) O} of every open set O R is open in X. ex Let f be a continuous function defined on a compact metric space. Use the characterization given in the previous exercise to prove that the range of f is compact. ex The range of a continuous map from a compact metric space into a metric space is compact. ex A uniformly continuous function f on [0, 1] maps Cauchy sequences to Cauchy sequences. ex Let f =(f 1,, f d ) be an R d -valued function on (X,ρ). a. Prove that f is continuous at a point x 0 X if and only if every component function f j, j = 1,...,d, is continuous at x 0 b. Prove that f is uniformly continuous on X if and only if every component function f j, j = 1,...,d, is uniformly continuous Equicontinuity. Let F be a set of continuous real-valued functions defined on a metric space (X,ρ). DEFINITION: F is equicontinuos at a point x X if for every ε > 0 there exist δ = δ(x,ε) such that if y B(x,δ) then for every f F, f (y) f (x) < ε. F is equicontinuous on X if it is equicontinuous at every x X.

14 2.5. Connectedness Connectedness DEFINITION: A metric space (X,ρ) is connected if it is not the union of two disjoint nonempty closed subsets. Since the complement of an open set is closed and the complement of a closed set is open, we can define connectedness by: (X,ρ) is connected if it is not the union of two disjoint nonempty open subsets. Proposition. An interval I R is connected. PROOF: Proof by contradiction. We consider first the case that I = [a,b] is bounded and closed. Assume I = E 1 E 2, with E 1 and E 2 nonempty and closed, numbered so that a E 1. Write c = infe 2. As [a,c) E 1, c is a limit point of E 1 and, as E 1 is closed, we have c E 1. On the other hand, being the infimum of the closed set E 2 we have c E 2, contradicting the assumption that E 1 and E 2 are disjoint. For a general interval J, that may be open, half open, unbounded, we assume J = E 1 E 2 with E j nonempty, closed and disjoint, take a E 1, b E 2, write I =[a,b],e j = E j I, and apply the case where I =[a,b] is bounded and closed.. Theorem. A continuous real-valued function defined on a connected space X (e.g. on [0,1]) has the intermediate value property, that is, if x 0,x 0 X, f (x 0 )=af(x 1 )=b and a < c < b, then there is y X such that f (y)=c. ex A closed subset E R is connected if and only if it is an interval. The interval can be bounded or extend to +, or to or to both. ex An interval is charecterized by the property: if a and b are points in it and a < c < b then c is in it as well. ex The range of a real-valued continuous function on a connected metric space is an interval. ex The range of a continuous map from a connected metric space into a metric space is connected.

15 24 2. Limits and convergence. 2.6 Series of functions Pointwise convergence. We now consider series sequences and series of (real-valued) functions defined on a finite interval, say on [0,1], or, more generally, on a metric space (X,ρ). The convergence of such a sequence or a series at a point x 0 is the convergence of a sequence or of a series of real numbers, which was studied earlier. Here we consider the Pointwise convergence of a sequence (or a series) of continuous functions, and the uniform convergence of such. Since the convergence of a series is defined by the convergence of the sequence of its partial sums (or, in the case of summability, of averages thereof) we focus on sequences. DEFINITION: The sequence { f n } converges pointwise on [0,1] if it converges at every point in the interval. We know that for real-valued functions convergence at a point x 0 is equivalent to the sequence { f n (x 0 )} being a Cauchy sequence, i.e., ε > 0, N = N(ε,x 0 ) such that if n,m > N then f n (x 0 ) f m (x 0 ) < ε. The sequence converges pointwise if the condition is satisfied for every x 0 [0,1] Uniform convergence. In the context and notation of the previous subsection, assume that { f n (x)} converges to f (x) everywhere on [0,1]. DEFINITION: The sequence { f n } converges to f uniformly means: ε > 0 N = N(ε) such that f n (x) f (x) ε for all x [0,1] and n > N. The uniformity is the fact that, for all ε > 0, there is a common N(ε) which accommodates all the points x [0,1]. Theorem. a. The sequence { f n } converges uniformly on on [0,1] if, and only if it is uniformly Cauchy, i.e., ε > 0, N = N(ε) such that for all x [0,1], if n,m > N then f n (x) f m (x) < ε. b. A uniform limit of a sequence { f n } of continuous functions is continuous. ex Assume f continuous on E and g(x) f (x) < ε for all x E. Prove: o g (x) < 2ε for all x E.

16 2.6. Series of functions Equicontinuity A set { f n } of real-valued functions on [0,1] is equicontinuous at a point x 0 if for every ε > 0 the exists δ(ε) > 0 such that if x 0 y < δ then f n (x 0 ) f n (y) < ε for all n. { f n } is uniformly equicontinuous if for every ε > 0 there exists δ(ε) such that if x y < δ then f n (x) f n (y) < ε for all n and for all x. The uniform continuity of a function is the fact that δ(ε) can be chosen so as to accommodate all x for the given function. The equicontinuity is the fact that δ(ε) can be chosen to accommodate all the functions f n (either at a specific point or at every point).

### PART I. THE REAL NUMBERS

PART I. THE REAL NUMBERS This material assumes that you are already familiar with the real number system and the representation of the real numbers as points on the real line. I.1. THE NATURAL NUMBERS

### Notes on metric spaces

Notes on metric spaces 1 Introduction The purpose of these notes is to quickly review some of the basic concepts from Real Analysis, Metric Spaces and some related results that will be used in this course.

### x if x 0, x if x < 0.

Chapter 3 Sequences In this chapter, we discuss sequences. We say what it means for a sequence to converge, and define the limit of a convergent sequence. We begin with some preliminary results about the

### CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e.

CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e. This chapter contains the beginnings of the most important, and probably the most subtle, notion in mathematical analysis, i.e.,

### Metric Spaces Joseph Muscat 2003 (Last revised May 2009)

1 Distance J Muscat 1 Metric Spaces Joseph Muscat 2003 (Last revised May 2009) (A revised and expanded version of these notes are now published by Springer.) 1 Distance A metric space can be thought of

### Mathematical Methods of Engineering Analysis

Mathematical Methods of Engineering Analysis Erhan Çinlar Robert J. Vanderbei February 2, 2000 Contents Sets and Functions 1 1 Sets................................... 1 Subsets.............................

### 1 if 1 x 0 1 if 0 x 1

Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or

### Undergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics

Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics An Introductory Single Variable Real Analysis: A Learning Approach through Problem Solving Marcel B. Finan c All Rights

### 1. Prove that the empty set is a subset of every set.

1. Prove that the empty set is a subset of every set. Basic Topology Written by Men-Gen Tsai email: b89902089@ntu.edu.tw Proof: For any element x of the empty set, x is also an element of every set since

### Mathematics for Econometrics, Fourth Edition

Mathematics for Econometrics, Fourth Edition Phoebus J. Dhrymes 1 July 2012 1 c Phoebus J. Dhrymes, 2012. Preliminary material; not to be cited or disseminated without the author s permission. 2 Contents

### God created the integers and the rest is the work of man. (Leopold Kronecker, in an after-dinner speech at a conference, Berlin, 1886)

Chapter 2 Numbers God created the integers and the rest is the work of man. (Leopold Kronecker, in an after-dinner speech at a conference, Berlin, 1886) God created the integers and the rest is the work

### {f 1 (U), U F} is an open cover of A. Since A is compact there is a finite subcover of A, {f 1 (U 1 ),...,f 1 (U n )}, {U 1,...

44 CHAPTER 4. CONTINUOUS FUNCTIONS In Calculus we often use arithmetic operations to generate new continuous functions from old ones. In a general metric space we don t have arithmetic, but much of it

### n k=1 k=0 1/k! = e. Example 6.4. The series 1/k 2 converges in R. Indeed, if s n = n then k=1 1/k, then s 2n s n = 1 n + 1 +...

6 Series We call a normed space (X, ) a Banach space provided that every Cauchy sequence (x n ) in X converges. For example, R with the norm = is an example of Banach space. Now let (x n ) be a sequence

### Metric Spaces. Chapter 7. 7.1. Metrics

Chapter 7 Metric Spaces A metric space is a set X that has a notion of the distance d(x, y) between every pair of points x, y X. The purpose of this chapter is to introduce metric spaces and give some

### Math212a1010 Lebesgue measure.

Math212a1010 Lebesgue measure. October 19, 2010 Today s lecture will be devoted to Lebesgue measure, a creation of Henri Lebesgue, in his thesis, one of the most famous theses in the history of mathematics.

### Chapter 1. Metric Spaces. Metric Spaces. Examples. Normed linear spaces

Chapter 1. Metric Spaces Metric Spaces MA222 David Preiss d.preiss@warwick.ac.uk Warwick University, Spring 2008/2009 Definitions. A metric on a set M is a function d : M M R such that for all x, y, z

### Ri and. i=1. S i N. and. R R i

The subset R of R n is a closed rectangle if there are n non-empty closed intervals {[a 1, b 1 ], [a 2, b 2 ],..., [a n, b n ]} so that R = [a 1, b 1 ] [a 2, b 2 ] [a n, b n ]. The subset R of R n is an

### Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011

Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011 A. Miller 1. Introduction. The definitions of metric space and topological space were developed in the early 1900 s, largely

### MA651 Topology. Lecture 6. Separation Axioms.

MA651 Topology. Lecture 6. Separation Axioms. This text is based on the following books: Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology by Nicolas Bourbaki Counterexamples

ADVANCED CALCULUS Lecture notes for MA 440/540 & 441/541 2015/16 Rudi Weikard log x 1 1 2 3 4 5 x 1 2 Based on lecture notes by G. Stolz and G. Weinstein Version of September 3, 2016 1 Contents First things

### No: 10 04. Bilkent University. Monotonic Extension. Farhad Husseinov. Discussion Papers. Department of Economics

No: 10 04 Bilkent University Monotonic Extension Farhad Husseinov Discussion Papers Department of Economics The Discussion Papers of the Department of Economics are intended to make the initial results

### Our goal first will be to define a product measure on A 1 A 2.

1. Tensor product of measures and Fubini theorem. Let (A j, Ω j, µ j ), j = 1, 2, be two measure spaces. Recall that the new σ -algebra A 1 A 2 with the unit element is the σ -algebra generated by the

### BANACH AND HILBERT SPACE REVIEW

BANACH AND HILBET SPACE EVIEW CHISTOPHE HEIL These notes will briefly review some basic concepts related to the theory of Banach and Hilbert spaces. We are not trying to give a complete development, but

### x a x 2 (1 + x 2 ) n.

Limits and continuity Suppose that we have a function f : R R. Let a R. We say that f(x) tends to the limit l as x tends to a; lim f(x) = l ; x a if, given any real number ɛ > 0, there exists a real number

### MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES Contents 1. Random variables and measurable functions 2. Cumulative distribution functions 3. Discrete

### Metric Spaces. Chapter 1

Chapter 1 Metric Spaces Many of the arguments you have seen in several variable calculus are almost identical to the corresponding arguments in one variable calculus, especially arguments concerning convergence

### Introduction to Topology

Introduction to Topology Tomoo Matsumura November 30, 2010 Contents 1 Topological spaces 3 1.1 Basis of a Topology......................................... 3 1.2 Comparing Topologies.......................................

### SOLUTIONS TO EXERCISES FOR. MATHEMATICS 205A Part 3. Spaces with special properties

SOLUTIONS TO EXERCISES FOR MATHEMATICS 205A Part 3 Fall 2008 III. Spaces with special properties III.1 : Compact spaces I Problems from Munkres, 26, pp. 170 172 3. Show that a finite union of compact subspaces

### So let us begin our quest to find the holy grail of real analysis.

1 Section 5.2 The Complete Ordered Field: Purpose of Section We present an axiomatic description of the real numbers as a complete ordered field. The axioms which describe the arithmetic of the real numbers

### University of Miskolc

University of Miskolc The Faculty of Mechanical Engineering and Information Science The role of the maximum operator in the theory of measurability and some applications PhD Thesis by Nutefe Kwami Agbeko

### INTRODUCTORY SET THEORY

M.Sc. program in mathematics INTRODUCTORY SET THEORY Katalin Károlyi Department of Applied Analysis, Eötvös Loránd University H-1088 Budapest, Múzeum krt. 6-8. CONTENTS 1. SETS Set, equal sets, subset,

### Discrete Mathematics: Solutions to Homework (12%) For each of the following sets, determine whether {2} is an element of that set.

Discrete Mathematics: Solutions to Homework 2 1. (12%) For each of the following sets, determine whether {2} is an element of that set. (a) {x R x is an integer greater than 1} (b) {x R x is the square

### Math 104: Introduction to Analysis

Math 104: Introduction to Analysis Evan Chen UC Berkeley Notes for the course MATH 104, instructed by Charles Pugh. 1 1 August 29, 2013 Hard: #22 in Chapter 1. Consider a pile of sand principle. You wish

### MEASURE AND INTEGRATION. Dietmar A. Salamon ETH Zürich

MEASURE AND INTEGRATION Dietmar A. Salamon ETH Zürich 12 May 2016 ii Preface This book is based on notes for the lecture course Measure and Integration held at ETH Zürich in the spring semester 2014. Prerequisites

### REAL ANALYSIS LECTURE NOTES: 1.4 OUTER MEASURE

REAL ANALYSIS LECTURE NOTES: 1.4 OUTER MEASURE CHRISTOPHER HEIL 1.4.1 Introduction We will expand on Section 1.4 of Folland s text, which covers abstract outer measures also called exterior measures).

### Automata and Rational Numbers

Automata and Rational Numbers Jeffrey Shallit School of Computer Science University of Waterloo Waterloo, Ontario N2L 3G1 Canada shallit@cs.uwaterloo.ca http://www.cs.uwaterloo.ca/~shallit 1/40 Representations

### FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES

FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES CHRISTOPHER HEIL 1. Cosets and the Quotient Space Any vector space is an abelian group under the operation of vector addition. So, if you are have studied

### The Lebesgue Measure and Integral

The Lebesgue Measure and Integral Mike Klaas April 12, 2003 Introduction The Riemann integral, defined as the limit of upper and lower sums, is the first example of an integral, and still holds the honour

### 9 More on differentiation

Tel Aviv University, 2013 Measure and category 75 9 More on differentiation 9a Finite Taylor expansion............... 75 9b Continuous and nowhere differentiable..... 78 9c Differentiable and nowhere monotone......

### Separation Properties for Locally Convex Cones

Journal of Convex Analysis Volume 9 (2002), No. 1, 301 307 Separation Properties for Locally Convex Cones Walter Roth Department of Mathematics, Universiti Brunei Darussalam, Gadong BE1410, Brunei Darussalam

### A Problem With The Rational Numbers

Solvability of Equations Solvability of Equations 1. In fields, linear equations ax + b = 0 are solvable. Solvability of Equations 1. In fields, linear equations ax + b = 0 are solvable. 2. Quadratic equations

### Cardinality. The set of all finite strings over the alphabet of lowercase letters is countable. The set of real numbers R is an uncountable set.

Section 2.5 Cardinality (another) Definition: The cardinality of a set A is equal to the cardinality of a set B, denoted A = B, if and only if there is a bijection from A to B. If there is an injection

### Open and Closed Sets

Open and Closed Sets Definition: A subset S of a metric space (X, d) is open if it contains an open ball about each of its points i.e., if x S : ɛ > 0 : B(x, ɛ) S. (1) Theorem: (O1) and X are open sets.

### FIRST YEAR CALCULUS. Chapter 7 CONTINUITY. It is a parabola, and we can draw this parabola without lifting our pencil from the paper.

FIRST YEAR CALCULUS WWLCHENW L c WWWL W L Chen, 1982, 2008. 2006. This chapter originates from material used by the author at Imperial College, University of London, between 1981 and 1990. It It is is

### Mathematics Review for MS Finance Students

Mathematics Review for MS Finance Students Anthony M. Marino Department of Finance and Business Economics Marshall School of Business Lecture 1: Introductory Material Sets The Real Number System Functions,

### P (A) = lim P (A) = N(A)/N,

1.1 Probability, Relative Frequency and Classical Definition. Probability is the study of random or non-deterministic experiments. Suppose an experiment can be repeated any number of times, so that we

### 2.3 Convex Constrained Optimization Problems

42 CHAPTER 2. FUNDAMENTAL CONCEPTS IN CONVEX OPTIMIZATION Theorem 15 Let f : R n R and h : R R. Consider g(x) = h(f(x)) for all x R n. The function g is convex if either of the following two conditions

### 6.254 : Game Theory with Engineering Applications Lecture 5: Existence of a Nash Equilibrium

6.254 : Game Theory with Engineering Applications Lecture 5: Existence of a Nash Equilibrium Asu Ozdaglar MIT February 18, 2010 1 Introduction Outline Pricing-Congestion Game Example Existence of a Mixed

### Convex analysis and profit/cost/support functions

CALIFORNIA INSTITUTE OF TECHNOLOGY Division of the Humanities and Social Sciences Convex analysis and profit/cost/support functions KC Border October 2004 Revised January 2009 Let A be a subset of R m

### 2.3 Scheduling jobs on identical parallel machines

2.3 Scheduling jobs on identical parallel machines There are jobs to be processed, and there are identical machines (running in parallel) to which each job may be assigned Each job = 1,,, must be processed

### Vector Spaces II: Finite Dimensional Linear Algebra 1

John Nachbar September 2, 2014 Vector Spaces II: Finite Dimensional Linear Algebra 1 1 Definitions and Basic Theorems. For basic properties and notation for R N, see the notes Vector Spaces I. Definition

### Chapter 7. Continuity

Chapter 7 Continuity There are many processes and eects that depends on certain set of variables in such a way that a small change in these variables acts as small change in the process. Changes of this

### THE BANACH CONTRACTION PRINCIPLE. Contents

THE BANACH CONTRACTION PRINCIPLE ALEX PONIECKI Abstract. This paper will study contractions of metric spaces. To do this, we will mainly use tools from topology. We will give some examples of contractions,

### DISINTEGRATION OF MEASURES

DISINTEGRTION OF MESURES BEN HES Definition 1. Let (, M, λ), (, N, µ) be sigma-finite measure spaces and let T : be a measurable map. (T, µ)-disintegration is a collection {λ y } y of measures on M such

### Measure Theory. Jesus Fernandez-Villaverde University of Pennsylvania

Measure Theory Jesus Fernandez-Villaverde University of Pennsylvania 1 Why Bother with Measure Theory? Kolmogorov (1933). Foundation of modern probability. Deals easily with: 1. Continuous versus discrete

### An Introduction to Real Analysis. John K. Hunter. Department of Mathematics, University of California at Davis

An Introduction to Real Analysis John K. Hunter Department of Mathematics, University of California at Davis Abstract. These are some notes on introductory real analysis. They cover the properties of the

### Lecture Note 1 Set and Probability Theory. MIT 14.30 Spring 2006 Herman Bennett

Lecture Note 1 Set and Probability Theory MIT 14.30 Spring 2006 Herman Bennett 1 Set Theory 1.1 Definitions and Theorems 1. Experiment: any action or process whose outcome is subject to uncertainty. 2.

### Probability and Statistics

CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 0 Probability and Statistics Kristel Van Steen, PhD 2 Montefiore Institute - Systems and Modeling GIGA - Bioinformatics ULg kristel.vansteen@ulg.ac.be

### Set theory as a foundation for mathematics

Set theory as a foundation for mathematics Waffle Mathcamp 2011 In school we are taught about numbers, but we never learn what numbers really are. We learn rules of arithmetic, but we never learn why these

### I. GROUPS: BASIC DEFINITIONS AND EXAMPLES

I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called

### Lecture Notes on Measure Theory and Functional Analysis

Lecture Notes on Measure Theory and Functional Analysis P. Cannarsa & T. D Aprile Dipartimento di Matematica Università di Roma Tor Vergata cannarsa@mat.uniroma2.it daprile@mat.uniroma2.it aa 2006/07 Contents

### Vector and Matrix Norms

Chapter 1 Vector and Matrix Norms 11 Vector Spaces Let F be a field (such as the real numbers, R, or complex numbers, C) with elements called scalars A Vector Space, V, over the field F is a non-empty

### E3: PROBABILITY AND STATISTICS lecture notes

E3: PROBABILITY AND STATISTICS lecture notes 2 Contents 1 PROBABILITY THEORY 7 1.1 Experiments and random events............................ 7 1.2 Certain event. Impossible event............................

### Lecture Notes on Topology for MAT3500/4500 following J. R. Munkres textbook. John Rognes

Lecture Notes on Topology for MAT3500/4500 following J. R. Munkres textbook John Rognes November 29th 2010 Contents Introduction v 1 Set Theory and Logic 1 1.1 ( 1) Fundamental Concepts..............................

### CHAPTER 5. Product Measures

54 CHAPTER 5 Product Measures Given two measure spaces, we may construct a natural measure on their Cartesian product; the prototype is the construction of Lebesgue measure on R 2 as the product of Lebesgue

### POWER SETS AND RELATIONS

POWER SETS AND RELATIONS L. MARIZZA A. BAILEY 1. The Power Set Now that we have defined sets as best we can, we can consider a sets of sets. If we were to assume nothing, except the existence of the empty

### Cartesian Products and Relations

Cartesian Products and Relations Definition (Cartesian product) If A and B are sets, the Cartesian product of A and B is the set A B = {(a, b) :(a A) and (b B)}. The following points are worth special

### TOPIC 4: DERIVATIVES

TOPIC 4: DERIVATIVES 1. The derivative of a function. Differentiation rules 1.1. The slope of a curve. The slope of a curve at a point P is a measure of the steepness of the curve. If Q is a point on the

### Point Set Topology. A. Topological Spaces and Continuous Maps

Point Set Topology A. Topological Spaces and Continuous Maps Definition 1.1 A topology on a set X is a collection T of subsets of X satisfying the following axioms: T 1.,X T. T2. {O α α I} T = α IO α T.

### Duality of linear conic problems

Duality of linear conic problems Alexander Shapiro and Arkadi Nemirovski Abstract It is well known that the optimal values of a linear programming problem and its dual are equal to each other if at least

### TOPIC 3: CONTINUITY OF FUNCTIONS

TOPIC 3: CONTINUITY OF FUNCTIONS. Absolute value We work in the field of real numbers, R. For the study of the properties of functions we need the concept of absolute value of a number. Definition.. Let

### CHAPTER 1 BASIC TOPOLOGY

CHAPTER 1 BASIC TOPOLOGY Topology, sometimes referred to as the mathematics of continuity, or rubber sheet geometry, or the theory of abstract topological spaces, is all of these, but, above all, it is

### Chapter 3. Cartesian Products and Relations. 3.1 Cartesian Products

Chapter 3 Cartesian Products and Relations The material in this chapter is the first real encounter with abstraction. Relations are very general thing they are a special type of subset. After introducing

### Linear Algebra I. Ronald van Luijk, 2012

Linear Algebra I Ronald van Luijk, 2012 With many parts from Linear Algebra I by Michael Stoll, 2007 Contents 1. Vector spaces 3 1.1. Examples 3 1.2. Fields 4 1.3. The field of complex numbers. 6 1.4.

### and s n (x) f(x) for all x and s.t. s n is measurable if f is. REAL ANALYSIS Measures. A (positive) measure on a measurable space

RAL ANALYSIS A survey of MA 641-643, UAB 1999-2000 M. Griesemer Throughout these notes m denotes Lebesgue measure. 1. Abstract Integration σ-algebras. A σ-algebra in X is a non-empty collection of subsets

### Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

### Elementary Number Theory We begin with a bit of elementary number theory, which is concerned

CONSTRUCTION OF THE FINITE FIELDS Z p S. R. DOTY Elementary Number Theory We begin with a bit of elementary number theory, which is concerned solely with questions about the set of integers Z = {0, ±1,

### 1 Norms and Vector Spaces

008.10.07.01 1 Norms and Vector Spaces Suppose we have a complex vector space V. A norm is a function f : V R which satisfies (i) f(x) 0 for all x V (ii) f(x + y) f(x) + f(y) for all x,y V (iii) f(λx)

### Foundations of Analysis. Joseph L. Taylor

Foundations of Analysis Joseph L. Taylor Version 2.5, Spring 2011 ii Contents Preface v 1 The Real Numbers 1 1.1 Sets and Functions.......................... 2 1.2 The Natural Numbers........................

### 10.2 Series and Convergence

10.2 Series and Convergence Write sums using sigma notation Find the partial sums of series and determine convergence or divergence of infinite series Find the N th partial sums of geometric series and

### INDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS

INDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS STEVEN P. LALLEY AND ANDREW NOBEL Abstract. It is shown that there are no consistent decision rules for the hypothesis testing problem

### INTRODUCTION TO TOPOLOGY

INTRODUCTION TO TOPOLOGY ALEX KÜRONYA In preparation January 24, 2010 Contents 1. Basic concepts 1 2. Constructing topologies 13 2.1. Subspace topology 13 2.2. Local properties 18 2.3. Product topology

### Extension of measure

1 Extension of measure Sayan Mukherjee Dynkin s π λ theorem We will soon need to define probability measures on infinite and possible uncountable sets, like the power set of the naturals. This is hard.

### THE PRIME NUMBER THEOREM

THE PRIME NUMBER THEOREM NIKOLAOS PATTAKOS. introduction In number theory, this Theorem describes the asymptotic distribution of the prime numbers. The Prime Number Theorem gives a general description

### Lemma 5.2. Let S be a set. (1) Let f and g be two permutations of S. Then the composition of f and g is a permutation of S.

Definition 51 Let S be a set bijection f : S S 5 Permutation groups A permutation of S is simply a Lemma 52 Let S be a set (1) Let f and g be two permutations of S Then the composition of f and g is a

### Metric Spaces. Lecture Notes and Exercises, Fall 2015. M.van den Berg

Metric Spaces Lecture Notes and Exercises, Fall 2015 M.van den Berg School of Mathematics University of Bristol BS8 1TW Bristol, UK mamvdb@bristol.ac.uk 1 Definition of a metric space. Let X be a set,

### What is Linear Programming?

Chapter 1 What is Linear Programming? An optimization problem usually has three essential ingredients: a variable vector x consisting of a set of unknowns to be determined, an objective function of x to

### The fundamental group of the Hawaiian earring is not free (International Journal of Algebra and Computation Vol. 2, No. 1 (1992), 33 37) Bart de Smit

The fundamental group of the Hawaiian earring is not free Bart de Smit The fundamental group of the Hawaiian earring is not free (International Journal of Algebra and Computation Vol. 2, No. 1 (1992),

### 6.2 Permutations continued

6.2 Permutations continued Theorem A permutation on a finite set A is either a cycle or can be expressed as a product (composition of disjoint cycles. Proof is by (strong induction on the number, r, of

### Chapter 1 - σ-algebras

Page 1 of 17 PROBABILITY 3 - Lecture Notes Chapter 1 - σ-algebras Let Ω be a set of outcomes. We denote by P(Ω) its power set, i.e. the collection of all the subsets of Ω. If the cardinality Ω of Ω is

### Columbia University in the City of New York New York, N.Y. 10027

Columbia University in the City of New York New York, N.Y. 10027 DEPARTMENT OF MATHEMATICS 508 Mathematics Building 2990 Broadway Fall Semester 2005 Professor Ioannis Karatzas W4061: MODERN ANALYSIS Description

### 2 Complex Functions and the Cauchy-Riemann Equations

2 Complex Functions and the Cauchy-Riemann Equations 2.1 Complex functions In one-variable calculus, we study functions f(x) of a real variable x. Likewise, in complex analysis, we study functions f(z)

### Real Roots of Univariate Polynomials with Real Coefficients

Real Roots of Univariate Polynomials with Real Coefficients mostly written by Christina Hewitt March 22, 2012 1 Introduction Polynomial equations are used throughout mathematics. When solving polynomials

### We give a basic overview of the mathematical background required for this course.

1 Background We give a basic overview of the mathematical background required for this course. 1.1 Set Theory We introduce some concepts from naive set theory (as opposed to axiomatic set theory). The

### Section 1.3 P 1 = 1 2. = 1 4 2 8. P n = 1 P 3 = Continuing in this fashion, it should seem reasonable that, for any n = 1, 2, 3,..., = 1 2 4.

Difference Equations to Differential Equations Section. The Sum of a Sequence This section considers the problem of adding together the terms of a sequence. Of course, this is a problem only if more than

### Elements of probability theory

2 Elements of probability theory Probability theory provides mathematical models for random phenomena, that is, phenomena which under repeated observations yield di erent outcomes that cannot be predicted

### INCIDENCE-BETWEENNESS GEOMETRY

INCIDENCE-BETWEENNESS GEOMETRY MATH 410, CSUSM. SPRING 2008. PROFESSOR AITKEN This document covers the geometry that can be developed with just the axioms related to incidence and betweenness. The full