Exercises from other sources REAL NUMBERS 2,...,

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1 Exercises from other sources REAL NUMBERS 1. Find the supremum and infimum of the following sets: a) {1, b) c) 12, 13, 14, }, { 1 3, 4 9, 13 27, 40 } 81,, { 2, 2 + 2, } 2,..., d) {n N : n 2 < 10}, e) {n/(m + n) : m, n N}, f) {n/(2n + 1) : n N}, g) {n/m : m, n N with m + n 10}. 2. Assume that S and T are two nonempty subsets of R such that T is bounded above and S T. Show that S is also bounded above and sup S sup T. 3. Let S and T be two nonempty subsets of R with the following property: for every x S and every y T we have x y. Show that S is bounded above, T is bounded below and sup S sup T. 4. (Nested Sequence Property) For each n N, assume we are given a closed interval I n = [a n, b n ] = {x R : a n x b n }. Assume also that each I n contains I n+1, i.e. I 1 I 2 I 3. Show that I n. Hint. Let S be the set of all a n s and T be the set of all b n s. Show that S and T have the property described in Question 3 and apply Question A real number x is called algebraic if there exist integers a 0, a 1, a 2,..., a n Z, not all zero, such that a n x n + a n 1 x n a 1 x + a 0 = 0. In other words, a real number is algebraic if it is the root of a polynomial with integer coefficients. For example 2 is algebraic as x = 2 satisfies the equation x 2 2 = 0 and the coefficients 1, 0, 2 of the polynomial x 2 2 are integers. Real numbers that are not algebraic are called transcendental numbers. a) Show that 3 2, are algebraic numbers.

2 b) Given n N (fixed) let X n be the set of all polynomials of degree n with integer coefficients. Show that X n is countable. c) Show that the set of all polynomials with integer coefficients is countable. d) Show that the set of all algebraic numbers is countable (Hint. A polynomial of degree n has at most n real roots.) e) What is the cardinality of the set of all transcendental numbers?

3 METRIC SPACES 6. Let X = R 2 and for p = (x 1, x 2 ), q = (y 1, y 2 ), { x2 + y d(p, q) = 2 + x 1 y 1 if x 1 y 1 x 2 y 2 if x 1 = y 1. Show that d is a metric on X. Illustrate by diagrams in the plane what the open balls are. 7. Let X be any set and d be the discrete metric. What are the open sets, what are the closed sets? 8. Let X = R 2, d = d 2, and E = {(x 1, x 2 ) : x 1 > x 2 }. Show that E is open in X. 9. Let E be a subset of a metric space. We define the boundary of E as E = E E. Show that a) Show that X = int(e) int(e ) E and the three sets on the right hand side are pairwise disjoint. b) E is closed E E. c) E is open E E =. 10. For the following sets E R 2, find E, E, int(e) and E. a) E = {(x, y) : 1 < x 2 + y 2 4}. b) E = {(x, y) : x > 0, xy = 1}. c) E = {(x, y) : x, y Q}. 11. Let X = R and E be as below. Decide whether E is open, closed, or neither. If E is not open, find a point of E which is not an interior point. If E is not closed, find a limit point of E which is not in E. a) E = Q, b) E = N, c) E = {x : x > 0}, d) E = {x : 0 < x 1}, e) E = {1 + 1/4 + 1/ /n 2 : n N}. 12. For a bounded nonempty set E X we define the diameter of E as Show that diam(e) = diam(e). diam(e) = sup{d(p, q) : p, q E}. 13. Let d be a metric on X. Show that ρ defined by ρ(p, q) = d(p, q) 1 + dp, q)

4 is also a metric on X. 14. Show that int(a B) = int(a) int(b) and int(a B) int(a) int(b).

5 COMPACTNESS AND CONNECTEDNESS 15. Identify which of the following sets are compact and which are not. If E is not compact, find an open cover of E which does not have any finite subcover. a) E = {(x, y) R 2 : 4 x 2 + y 2 9}, b) E = {(x, y) R 2 : 4 x 2 + y 2 < 9}, c) E = {(x, y) R 2 : 0 < x 1 and y = sin(1/x)}, d) E = {(x, y) R 2 : xy 1}. 16. Given two non-empty subsets A and B of a metric space, we define the distance between them as dist(a, B) = inf{d(x, y) : x A, y B}. Assume further that B is compact. Show the following: dist(a, B) = 0 A B. 17. Let A be a nonempty compact set of a metric space X and b be a point in X. Show that there is a point a A such that dist(a, {b}) = d(a, b). 18. For a nonempty, bounded subset A of a metric space, we define the diameter of A as diam(a) = sup{d(x, y) : x, y A}. Show that if A is compact, then there are points x 0, y 0 A such that diam(a) = d(x 0, y 0 ). 19. Let A and B two nonempty, compact subsets of a metric space such that A B =. a) Show that dist(a, B) > 0. b) Show that there are two open sets U and V such that A U, B V and U V =. 20. Let X be a compact metric space (i.e. X is a compact subset of itself). Show that there is a subset A of X with the following properties: i) A is at most countable, and ii) A = X. 21. Identify which of the following sets are connected and which are not. If E is not connected, find two nonempty separated sets A and B such that E = A B. a) In X = R, E = Q, b) In X = R 2, E = B 1 (p 1 ) B 1 (p 2 ) {(x, 0) : 1 < x < 1}, where p 1 = ( 2, 0), p 2 = (2, 0), c) In X = R 2, E = {(x, y) : 0 < y x 2, x 0} {(0, 0)}, d) In X = R 2, E = {(x, y) : y = sin(1/x), x 0} {(0, y) : 1 y 1}. 22. Show that the intersection of two connected sets in R is connected. Show that this is false if R is replaced by R 2.

6 23. Show that if E R is connected, then int(e) is also connected. Show that this is false if R is replaced by R Let E be a connected subset of a metric space X, and let A be any subset of X such that E A E. Show that A is also connected.

7 SEQUENCES 25. Show that the sequence {p n } in R 2 defined by p n = (n, 1/n) does not converge. ( 26. Show that the sequence {p n } in R 2 defined by p n = n+1 n ), ( 1)n converges. n 27. Let {p n } be a sequence and p be a point in a metric space. Assume for all n we have d(p n+1, p) 1 2 d(p n, p). Show that lim n p n = p. 28. Among the following sequences, some are subsequences of others; determine all those which are so related. a) {1, 1, 1, 1,...}, b) {1, 1, 1, 1, 1, 1,...}, c) {1, 1, 1, 1, 1,...}, d) {1, 1, 1, 1, 1,...}, e) {1, 0, 1, 0, 1, 0, 1, 0,...} Let {x n }, {a n } and {b n } be three sequences in R such that a n x n b n for every n. Assume also lim n a n = L and lim n b n = L. Show that lim n x n = L also (this property is sometimes called Sandwich Property). 30. What are the correct hypotheses for the following statement? If lim n x n = A, then lim n xn = A. Prove the above statement. 31. Find the lim sup and lim inf of the following sequences ({x n } in R: a) x n = ( 1) n, b) x n = ( 1) n ), n c) x n = n + ( 1)n (2n + 1) (, d) x n = sin n π ). n 3

8 SERIES 32. Show that if a n converges, then n=n a n 0 as N. 33. Investigate the convergence of a n where n n + 1 n + 1 n a) a n = b) a n = n + 1 n + 1 c) d) e) 1 ( 2 ( ) 3 ( ) ) f) a n where 3 3 a n = ( ) 1 3 3n/2 7 7 n 2 3n/2 3 n if n is even if n is odd 34. Let a n and b n converge, with a n > 0, b n > 0 for all n. Suppose that a n /b n L as n. Prove that / a n b n L as N. n=n n=n 35. Some of the following statements are true and some are false; prove those that are true, and prove those that are false. a) If a n and b n converge, then (a n + b n ) also converges. b) If a n and b n diverge, then (a n + b n ) also diverges. c) If a n diverges, then (a n ) 2 also diverges. d) If a n is convergent, then (a n ) 2 is also convergent. e) If a n and b n converge, then a n b n also converges. f) If (a n ) 2 converges, then a n /n also converges. g) If 0... < a n+1 < a n <... < a 2 < a 1, lim n a n = 0 and a n is convergent, then lim n na n = Show that if f 0 and f is decreasing, and if c n = n f(k) n k=1 1 f(x) dx, then lim n c n exists. 37. If p and q are fixed integers with 1 p q, show that lim qn n k=pn 1 k = ln q p.

9 38. Find the radius of convergence of the following power series: n! a) n n xn, b) c n2 x n where c > 0 constant, n=0 x n c) n, d) (2n 1) x n (2n)

10 CONTINUITY 39. Let (X, d) be a discrete metric space and Y be an arbitrary metric space. Show that every function f : X Y is continuous. 40. Let X and Y be metric spaces, and f : X Y. Show that a) f is continuous on X if and only if f(a) f(a) for every subset A of X. b) f is continuous on X if and only if f 1 (int(b)) int(f 1 (B)) for every subset B of Y. c) f is continuous on X if and only if f 1 (B) f 1 (B) for every subset B of Y. 41. Let f : [0, 1] R be a continuous function and for 0 s 1 we define g f (s) = s 0 f(t)dt. a) Show that g f : [0, 1] R is also continuous (i.e. whenever f C[0, 1] we have that g f C[0, 1]. b) Show that the function F : C[0, 1] C[0, 1] defined by F (f) = g f is uniformly continuous on C[0, 1] (Recall that the metric of C[0, 1] is d(ϕ, ψ) = sup{ ϕ(t) ψ(t) : 0 t 1}.) 42. Let X, d) be a compact metric space, and f : X X be a function such that d(f(u), f(v)) = d(u, v) for all pairs of points u, v X. Then show that f is onto. 43. Let M be the subset of C[0, 1] consisting of all continuous functions f : [0, 1] R such that f(0) = 0, f(1) = 1, and sup{ f(t) : 0 t 1} 1. Show that the function is a continuous function of M into R. F (f) = 1 0 (f(t) 2 ) dt 44. Let (X, d) be a metric space and f : X X be continuous. a) Let g : X R be defined by g(x) = d(f(x), x). Show that g is also continuous. b) Assume further that X is compact and f(x) x for all x X. Show that there is a strictly positive constant c such that d(f(x), x) c for all x X.

11 SEQUENCES AND SERIES OF FUNCTIONS 45. Which of the following sequences of functions converge uniformly on [0, 1]? a) f n (x) = x/(1 + nx), b) f n (x) = nxe nx2, c) f n (x) = nx(1 x) n, d) f n (x) = nx(1 x 2 ) n2, e) f n (x) = x n /(1 + x n ), f) f n (x) = n x x n cos nx, g) f n (x) = nx(1 x) n, h) f n (x) = xe nx2, i) f n (x) = n 2 xe nx. 46. Let g be continuous on [0, 1] with g(1) = 0. Show that the sequence {f n } where f n (x) = g(x)x n converges uniformly on [0, 1]. 47. Let {f n } be a sequence of continuous real valued functions defined on [a, b], let {a n }, {b n } be sequences in [a, b] with lim n an = a, lim n bn = b. Suppose that {f n } converges uniformly to f on [a, b]. Show that bn lim f n (x) dx = n a n b a f(x) dx. 48. Let X, d) be a metric space, f n : X R a sequence of functions that are uniformly continuous on X, which converges uniformly to a function f : X R. Show that f is also uniformly continuous on X. 49. Find an interval I R on which the following series of functions is uniformly convergent. ( x ) n, sin 2 nx a) b) 3 2 n 1, x c) (1 + x), d) x n [1 + (n 1)x][1 + nx]. 50. Show that the following series is uniformly convergent on E = [0, ). x n + n 2 x Find the sum of the series 2 n n 2 3 n.

12 52. a) Let f(x) = sin nx π. Show that f(x) dx = 2 n Show that the following limits exist and evaluate them. 1 3 a) lim e x2 /n dx, b) lim n 1 n 0 π/2 c) lim n 0 ( x sin n) ( x + cos dx. n) 1 (2n 1) 4. nx x 3 + nx dx,