MATH & MATH INTRODUCTION TO ANALYSIS EXERCISES FALL 2016 & SPRING Scientia Imperii Decus et Tutamen 1

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1 MATH & MATH INTRODUCTION TO ANALYSIS EXERCISES FALL 2016 & SPRING 2017 Scientia Imperii Decus et Tutamen 1 Robert R. Kallman University of North Texas Department of Mathematics 1155 Union Circle Denton, Texas office telephone: ; office fax: ; office kallman@unt.edu August 17, Taken from the coat of arms of Imperial College London.

2 Fall 2016 Exercise 1. Prove that A = A, A B = B A, A c B c = A B, (A B) c = A c B = A B c, A (B C) = (A B) (A C), and (A B) C = A (B C). This proves that the set of subsets of a given set U form a commutative ring with identity with * = and + =. Exercise 2. (a, b) = (c, d) if and only if a = c and b = d. Exercise 3. A relation R X X with domain(r) = X is an equivalence relation if and only if R = R 1 = R R. Exercise 4. Let = E R and let s R. Then s = sup(e) = lub(e) if and only if: (i) x s for all x E; (ii) for each ϵ > 0, there exists x E such that x > s ϵ; and s = inf(e) = glb(e) if and only if: (iii) x s for all x E; (iv) for each ϵ > 0, there exists x E such that x < s + ϵ. Exercise 5 (Principle of Strong Induction). Let S N have the following two properties: (i) 1 S; (ii) m + 1 S if m N satisfies r S for all r N with r m. Then S = N. Hint: Use the Well-Ordering Property of N. Exercise 6. Prove the following slight but useful generalization of the Recursion Theorem: Let S be a set, a S, and φ : N S S. Then there exists one and only one function f : N S such that f(1) = a and f(n + 1) = φ(n, f(n)). Exercise 7. Prove the Laws of Exponents. Hint: Fix x, y, and m and induct on n, considering the cases n = 0, n > 0, and n < 0 separately. 1

3 Exercise 8. Let 0 < λ < 1. Let E = {λ n n N}. Prove that glb(e) = 0. Exercise 9. Define s 1 = 1/2 and inductively define s n+1 = s n + 1/2 n+1. Let E = {s n n N}. Find lub(e). Exercise 10. Let x R. Then x = lub({q Q q < x}) = lub({q Q q x}) = glb({q Q q > x}) = glb({q Q q x}). Exercise 11. If x R, let A = {q Q q x} and let B = {q Q q > x}. Prove that the pair A, B forms a Dedekind cut of Q. The same is true if A = {q Q q < x} and B = {q Q q x}. Note that these two cases are the same if x is irrational. Prove that if A, B is a Dedekind cut of R, then A = A Q and B = B Q is a Dedekind cut of Q. Finally, prove that if A, B is a Dedekind cut of Q, then there is a unique x R such that the pair A, B has one of the two forms listed above. This exercise gives a hint on how to construct R from Q. This exercise is the basis for the theorem that any two ordered fields which are Dedekind complete, i.e., with the property that any nonempty set with an upper bound has a least upper bound, are isomorphic. Thus there is at most one object satisfying Axioms I - VIII. Exercise 12. Let F 1, F 2 be fields and φ : F 1 F 2 satisfy φ(a + b) = φ(a) + φ(b) and φ(ab) = φ(a)φ(b) for all a, b F 1. Prove that if there is some a 0 F 1 such that φ(a 0 ) 0, then φ is one-to-one, φ(1) = 1, φ( a) = φ(a), φ(a 1 ) = φ(a) 1 for all 0 a F 1, and φ(f 1 ) is a subfield of F 2. If φ(f 1 ) = F 2, then φ is said to be a (field) isomorphism. If F 1 = F 2 and φ(f 1 ) = F 1, then φ is said to be a (field) automorphism. Let φ : R F R be a field isomorphism. Then F = R and φ is the identity. In particular, every field automorphism of R is the identity. Exercise 13. C cannot be ordered, i.e., there is no possible choice of P C such that Axiom VI and Axiom VII hold with R replaced by C. Hint: i P or i P. Exercise 14. [ ] a b Prove that the set F of 2 2 matrices of the form, where a, b R, is a field. Prove [ ] b a a b that the mapping φ : C F, φ : (a, b), onto F, is a field isomorphism. Prove b a 2

4 that φ is an isomorphism of R with the set of 2 2 diagonal matrices and that φ(z) = φ(z) t. What is det(φ(a, b))? Exercise 15. [ ] w z Consider the set H of 2 2 matrices with complex entries of the form. The elements z w of H are called quaternions. Prove that H is closed under matrix addition and multiplication. Prove that the set of diagonal quaternions is isomorphic to the field C[ and that ] the adjoint w z operation extends the complex conjugation operation. If q H, q = 0, find an z w explicit expression for det(q) and for q 1. Check that H satisfies all of Axioms I - V except that multiplication is not commutative. This proves that H forms a noncommutative division ring. Exercise 16. Let x 1 and let E = { n x n N}. Prove that 1 = glb(e). Hint: Let ϵ > 0. Use Bernoulli s Inequality to prove that there is some n N such that x < (1 + ϵ) n. Next, if 0 < x 1 and E = { n x n N}, prove that 1 = lub(e). Exercise 17 (Chebyshev s Inequality). Let a 1 a 2 a n and b 1 b 2 b n be real numbers. Prove that ( n i=1 a i)( n k=1 b k) n n j=1 a jb j and that we have equality if and only if either a 1 = a n or b 1 = b n. Hint: First prove that 0 n n i=1 k=1 (a i a k )(b i b k ). Exercise 18. If a 1, a 2,..., a n are all positive, then ( n j=1 a j)( n j=1 1 a j ) n 2 and we have equality if and only if a 1 = a 2 = = a n. If a, b, and c are positive and a + b + c = 1, then (1/a 1)(1/b 1)(1/c 1) 8 and we have equality if and only if a = b = c = 1/3. Exercise 19. If a 1, a 2,..., a n are positive real numbers, then n a 1 a 2 an n a 1 a 2... a n. When does equality hold in this inequality? The left hand side of this inequality is called the harmonic mean of a 1, a 2,..., a n. Exercise 20. Let A be a nonempty set and let X = {0, 1}. Prove that there exits a bijection f : P(A) X A. 3

5 Exercise 21. Finite Axiom of Choice: Given a finite collection A of pairwise disjoint nonempty sets, there exists a set C having exactly one element in common with each element of A, i.e, such that for each A A, the set C A contains a single element. Prove the Finite Axiom of Choice without using the Axiom of Choice. Exercise 22. Let I be a finite set and for each ι I let A ι be a finite set. Then ι I A ι and ι I A ι are finite sets. Exercise 23. Let A be an infinite set and let B A be a finite subset. Then A B is infinite. Let A be an uncountable set and let B A be a countable subset. Then A B is uncountable. Exercise 24. Let A be an infinite set and let B be a countable set. Then A A B. Exercise 25. Let A be an uncountable set and let B A be a countable subset. Then A B A. Exercise 26. Let A and B be sets. Then A B A B if both A and B have at least two distinct elements. Exercise 27. Let a, b R, a < b. Prove that [a, b and a, b] are neither open nor closed. Exercise 28. Let X be a set, let d 1 and d 2 be two metrics on X, and let T 1 and T 2 be the corresponding topologies on X. We say that d 1 and d 2 are (topologically) equivalent if and only if T 1 = T 2. This notion of (topological) equivalence is an equivalence relation on the set of metrics on X. Note that d 1 and d 2 are (topologically) equivalent if and only if T 1 = T 2 T 1 T 2 and T 2 T 1. Prove that T 1 T 2 if and only if the following condition holds: ( ) if a X and δ > 0, then there exists ϵ > 0 such that B d2 (a, ϵ) B d1 (a, δ). Prove that if T 1 T 2, then (X, d 1 ) is compact if (X, d 2 ) is compact. Suppose that there exists C > 0 such that d 1 (x, y) Cd 2 (x, y) for all x, y X. Prove that T 1 T 2. 4

6 Let (X, d), (Y, ρ) be metric spaces. Define functions d 1, d 2, and d 3 : (X Y ) (X Y ) [0, + by d 1 ((x 1, y 1 ), (x 2, y 2 )) = d(x 1, x 2 )+ρ(y 1, y 2 ), d 2 ((x 1, y 1 ), (x 2, y 2 )) = max(d(x 1, x 2 ), ρ(y 1, y 2 )), and d 3 ((x 1, y 1 ), (x 2, y 2 )) = d(x 1, x 2 ) 2 + ρ(y 1, y 2 ) 2. Prove that d 1, d 2, and d 3 are (topologically) equivalent metrics. Also prove that B d2 ((x, y), δ) = B d (x, δ) B ρ (y, δ). Exercise 29. Let a, b, c [0, + be such that a b + c. Prove that a b + c 1+a 1+b d(x,y) 1+c 1+d(x,y). Let (X, d) be a metric space. Define a function ρ : X X [0, + by ρ(x, y) =. Prove that ρ is a metric on X which is equivalent to d and that X is bounded with respect to ρ. The same is true if ρ(x, y) = min(d(x, y), 1). Exercise 30. Let (X, d) be a metric space and = A X. The diameter of A, denoted diam(a), is defined to be lub{d(a 1, a 2 ) a 1, a 2 A}. Prove that diam(a) < + if and only if A is bounded. Exercise 31. Let C > 0 and a R n. Prove that B = {x R n x a C} is compact. Hint: prove that B is closed and is a subset of a compact set. Exercise 32 (Squeeze Theorem or Pinching Theorem or Sandwich Theorem). Let p Z and let (a n ) + n=p, (b n ) + n=p and (c n ) + n=p be three sequences in R such that a n b n c n for all n p and such that a n L and c n L as n +. Then b n L as n +. Exercise 33. Let (X, d) and (Y, ρ) be complete metric spaces and let d 1, d 2, and d 3 be the metrics defined on X Y in Exercise 28. Then X Y is complete with repect to d 1, d 2, and d 3. Exercise 34. If two series eventually agree, then they either both converge or both diverge. Exercise 35. Compute n 1 1/[(n + 1) n + n n + 1]. Hint: Rationalize denominators and telescope. Exercise 36. If m, n N, then 1 + m k=1 and induction. ( n+k k ) = ( n+m+1 m ). Hint: Write 1 as ( n+1 0 ), use Proposition??, Exercise 37. Suppose that (X, d) is a metric space and A Y X. Then cl Y (A) = cl X (A) Y. 5

7 Exercise 38. Let X be a set and d 1 and d 2 two metrics on X with corresponding topologies T 1 and T 2. Prove that T 1 T 2 if and only if the following condition holds: ( ) if {x n } n p X is a sequence, x X and d 2 (x n, x) 0 as n +, then d 1 (x n, x) 0 as n +. Exercise 39. Let (X, d) be a complete metric space and A a totally bounded subset of X. Then cl X (A) is compact. 6

8 Spring 2017 Exercise 40. Let (x n ) + n=p R #. If x R # is the limit of a subsequence of (x n ) + n=p, then lim inf n + x n x lim sup n + x n. Prove that both lim sup n + x n and lim inf n + x n are limits of a subsequence of (x n ) + n=p. Exercise 41. Let (x n ) + n=p R. If x R is a limit point of the elements of the sequence (x n ) + lim inf n + x n x lim sup n + x n. n=p, then Exercise 42. Let (x n ) + n=p and (y n ) + n=p be sequences in R # such that x n y n for all n N p. Then lim inf n + x n lim inf n + y n and lim sup n + x n lim sup n + y n. Exercise 43. Let (x n ) + n=p R #. Then lim sup n + x n = lim inf n + ( x n ) and lim inf n + x n = lim sup n + ( x n ). Exercise 44. Let L R and (x n ) + n=p, (y n ) + n=p [L, + ] R #. Then lim sup n + (x n +y n ) lim sup n + x n + lim sup n + y n. Exercise 45. Prove that l/ l l! e as l +. Hint: consider al = l l /l!. Exercise 46. cos(z) = z2l l 0 ( 1)l, sin(z) = z2l+1 (2l)! l 0 ( 1)l, cos(z) = cos( z), sin( z) = sin(z), (2l+1)! exp(iz) = cos(z) + i sin(z), and exp( iz) = cos(z) i sin(z) for all z C. Furthermore exp(0) = 1, exp(1) = e, cos(0) = 1 and sin(0) = 0. Exercise 47. Let (c n ) + n=1 [0, + satisfy 0 < c n+1 c n for all n 1 and n 1 c n < +. Prove there exist two divergent series n 1 a n and n 1 b n so that a 1 a 2 a 3 > 0, b 1 b 2 b 3 > 0 and c n = min(a n, b n ) for all n 1. Exercise 48. Suppose that n 0 a n and n 0 b n are two absolutely convergent series and that n 0 c n is their Cauchy product. Then n 0 c n is absolutely convergent, n 0 c n ( n 0 a n )( n 0 b n ), and n 0 c n = ( n 0 a n)( n 0 b n). 7

9 Exercise 49. (z n ) + n=p C, z C, z n z = z n z. w, z C = exp(z) 0, exp( z) = 1/ exp(z), exp(z) = exp(z), cos(z) = cos(z), sin(z) = sin(z), cos 2 (z) + sin 2 (z) = 1, cos(w + z) = cos(w) cos(z) sin(w) sin(z), cos(w z) = cos(w) cos(z) + sin(w) sin(z), sin(w + z) = sin(w) cos(z) + cos(w) sin(z), sin(w z) = sin(w) cos(z) cos(w) sin(z), sin(2z) = 2 sin(z) cos(z), and cos(2z) = cos 2 (z) sin 2 (z) = 2 cos 2 (z) 1 = 1 2 sin 2 (z). n Z = (exp(z)) n = exp(nz). t R = exp(t) > 0, cos(t) and sin(t) R, and exp(it) = 1. Furthermore s, t R and s < t = exp(s) < exp(t) (i.e., exp is strictly increasing on R), cos(t) 1 and sin(t) 1. Finally, n Z and z C = (cos(z) + i sin(z)) n = cos(nz) + i sin(nz) (de Moivres Theorem or de Moivres Formula). Exercise 50. Let X be a set and n 1. If f BF(X, C n ), define f = sup x X f(x) [0, + >. f is called the (uniform) norm of f. If f, g BF(X, C n ) and λ C, prove that f + g f + g, f = 0 f = 0, and λf = λ f. If f, g BF(X, C n ) define d(f, g) = f g. Prove that d is a metric on BF(X, C n ). Sequences convergent with respect this metric are said to be uniformly convergent. Prove that the mapping f f, BF(X, C n ) R, is uniformly continuous. Exercise 51. If (X, d) is a metric space and n 1, then CF(X, C n ) and BCF(X, C n ) are vector spaces over C and CF(X, R n ) and BCF(X, R n ) are vector spaces over R. Furthermore, CF(X, C) and BCF(X, C) are algebras over C and CF(X, R) and BCF(X, R) are algebras over R. Finally, BF(X, C n ) is a complete metric space and BCF(X, C n ) and BCF(X, R n ) are closed topological subspaces and therefore are themselves complete. Hint: Use the fact that C n is a complete metric space and Theorem??. Exercise 52. If (X, d) is a metric space, A X is connected and A C cl X (A) X, then C is connected. Exercise 53. Let X be a set, let d and ρ be two metrics on X with corresponding topologies T d and T ρ and let i : (X, d) (X, ρ), i : x x be the natural identification mapping. Then T ρ T d if and only if i is continuous. Furthermore d and ρ are equivalent metrics if and only if i is a homeomorphism. 8

10 Exercise 54. If I R is an interval and f : I R is continous and one-to-one, then f is strictly monotone on I. Exercise 55. Let f : [a, b] [c, d] be monotone, one-to-one and onto. Prove that f is continuous. Exercise 56. Let a, b and c C. Then: (i) a b+c = a b a c if a > 0; (ii) (ab) c = a c b c if a > 0 and b > 0; (iii) a b > 0 if a > 0 and b R, in which case log(a b ) = b log(a); (iv) (a b ) c = a bc if a > 0 and b R. Exercise 57. If a, b > 0, prove the following statements: (i) lim x + x a / exp(bx) = 0; (ii) lim x + log(x)/x a = 0; (iii) lim x 0 x a log(x) = 0. Exercise 58. If a, b R and a > 0, prove the following statements: (i) lim h 0 ((a h 1)/h) = log(a) (h C); (ii) lim h 0 ((a z+h a z )/h) = log(a)a z (z, h C); (iii) lim n + n( n a 1) = log(a); (iv) lim h 0 (1 + bh) 1/h = e b (h R); (v) lim n + (1 + b/n) n = e b. Exercise 59. Calculate D(x x ) (x > 0). Exercise 60. Let a > 0, a 1, b R, x > 0, and y > 0. Then: (1) log a (1) = 0; (2) log a (a) = 1; (3) log e (x) = log(x); (4) log a (xy) = log a (x) + log a (y); (5) log a (1/x) = log a (x); (6) log a (x/y) = log a (x) log a (y); 9

11 (7) log a (x b ) = b log a (x); (8) (log a ) (x) = 1/x log(a); (9) (d/dx)(log x (a)) = log(a)/x log 2 (x); (10) a log a (b) = b if b > 0; (11) log a (a x ) = x Exercise 61. Generalize Bernoulli s Inequality (Theorem??) as follows. Let α R, 0 α 1, be fixed. Then x R, x > 1, x 0 = (1 + x) α > 1 + αx if α < 0 or α > 1 and (1 + x) α < 1 + αx if 0 < α < 1. Exercise 62. Prove that sin(2π/12) = sin(π/6) = cos(π/3) = cos(2π/6) = 1/2 and that sin(2π/6) = sin(π/3) = cos(π/6) = cos(2π/12) = 3/2. Exercise 63. cos(2π/10) = cos(π/5) = (1 + 5)/4. Hint: Let θ = π/10 and find a cubic equation satisfied by cos(2θ). Conclude that cos(2π/5) = ( 1 + 5)/4. Exercise 64. Prove that C (R) and C 0 (R) are algebras over R which are closed under translations (T a (φ)(x) = φ(x + a)), scalings (S λ (φ)(x) = φ(λx), λ 0), and differentiation. C (R) is closed under composition but C 0 (R) is not. Let P (x) be a polynomial with coefficients in R. Then lim t 0 P (1/t) exp( 1/t) = 0. Let ψ(t) = 0 if t 0 and let ψ(t) = exp( 1/t) if t > 0. Then ψ C (R). Define φ(x) = ψ(1 x 2 ) for x R. Then φ(0) = 1/e > 0, φ 0, and φ C 0 (R) (actually φ 0 outside of [ 1, 1]). Exercise 65. Evaluate lim x + (e x + x) 1/x. Exercise 66. Evaluate lim x 1 [ x x 1 1 log(x) ]. Exercise 67. Let I R be an interval and let f, g : I C, λ C, t I, and (Df)(t) and (Dg)(t) exist. Then D(λf)(t), D(fg)(t) and D(f/g)(t) (if g(t) 0) exist and D(λf)(t) = λd(f)(t), (D(fg))(t) = (Df)(t)g(t) + f(t)(dg)(t) and D(f/g)(t) = [(Df)(t)g(t) f(t)(dg)(t)]/g(t) 2 provided g(t) 0. 10

12 Exercise 68. Consider the statements: (i) f R(α, [a, b]); (ii) there exists L R such that for every ϵ > 0, there exists a δ > 0 such that if P = {a = x 0 x 1 x n = b} is any partition of [a, b] with P < δ = L 1 l n f(τ l) α l < ϵ for all choices of τ l [x l 1, x l ]. Then (ii) = (i), in which case L = b a (ii) is true if α is continuous, in which case L = b a fdα. Exercise 69. Prove fdα, (i) = (ii) is false in general, and (i) = Theorem 1 (Bliss Theorem). Let [a, b] R 1 be a closed bounded interval, k 1 a positive integer, ψ a continuous function R-valued function on the cube [a, b] k, and P = {a = t 0 < t 1 < < t n 1 < t n = b} a partition of [a, b]. Let ϕ(t) = ψ(t,..., t). ϕ : [a, b] R 1 is a continuous function. Sums of the form 1 l n ψ(τ l,1,..., τ l,k )(t l t l 1 ), where (τ l,1,..., τ l,k ) [t l 1, t l ] k is arbitrary, converge to b ϕ(t)dt as P 0. a 11