EXAM II: PRACTICE SHEET (1) All the HW problems! (2) (a) Let (X, d) be a metric space. Prove that the closed neighborhood N r (x) = {y X : d(x, y) r} is a closed set. (b) Let (Rn , d) with the standard metric. Prove that the closure of the open neighborhood Nr (x) is the closed neighborhood. (c) Is it true in general that the closure of the open neighborhood Nr (x) is the closed neighborhood? If true, prove it and if false, give a counter example? (3) Let X be a metric space. For any two nonempty subsets A, B X define d(A, B) = inf{d(x, y) : x A, y B}. Note that if A B 6= , then d(A, B) = 0. Prove or provide a counter to the following statement. (a) If A and B are two disjoint subsets of X, then d(A, B) > 0. (b) If A and B are two separated subsets of X, then d(A, B) > 0. (c) If A and B are two disjoint open subsets of X, then d(A, B) > 0. (d) If A and B are two disjoint closed subsets of X, then d(A, B) > 0. (e) If A and B are two disjoint compact subsets of X, then d(A, B) > 0. (4) Show that every closed subset of Rn is an intersection of countably many open sets. (5) Let {x1 , x2 , . . .} be a countable subset of R. Assume that there are finitely many point y1 , . . . , ym R so that the following holds. \u000f > 0, N such that n > N, 1 ` m, |xn y` | < \u000f. Set K = {x1 , x2 , . . .} {y1 , . . . , ym }. (a) Prove that K is compact. (b) Using the definition of a compact set, prove that K is compact. (In part (b) you are supposed to use the definition and not the Heine-Borel theorem.) (6) Rudin, Chapter 2 (page 43), problem 30. (7) Prove that a metric space X is connected if and only if the following holds. The only subsets of X which are both closed and open are and X. (8) Let A Rn . Recall that a point x Rn is called a condensation point of A if Nr (x) A is uncountable for any neighborhood Nr (x). Prove that if A is uncountable, then there exists some x A which is a condensation point of A. (9) Let a R, prove that limn an n! 1 = 0. 2 EXAM II: PRACTICE SHEET (10) Let {xn } be a sequence in R with infinitely many distinct terms. Prove that {xn } either has an increasing subsequence or a decreasing subsequence. (11) Define dm on R2 as follows. \u0010 \u0011 dm (x1 , x2 ), (y1 , y2 ) = max{|x1 y1 |, |x2 y2 |}. (a) Prove that dm is a metric on R2 . (b) Describe Nrm (x) = {y R2 : dm (x, y) < r}. (c) Let d denote the standard metric on R2 . Prove that xn x in (R2 , d) if and only if xn x in dm . (d) Prove that A R2 is an open subset of (R2 , d) if and only if it is an open subset of (R2 , dm ). (e) Prove that A R2 is a closed subset of (R2 , d) if and only if it is a closed subset of (R2 , dm ). (f) Prove that A R2 is a compact subset of (R2 , d) if and only if it is a compact subset of (R2 , dm ). P an P (12) Let an 0 for all n N and assume that a2n converges. Prove that n converges. (13) Let X = {(a1 , a2 , . . .) : an R and |an | 1 for all n N}. Define a distance on X as follows. \u0001 d (an ), (bn ) = sup{|an bn | : n N}. (a) Prove that X is not a separable metric space. (b) Note that X is bounded and closed. Is X compact