Coursehero >> Colorado >> Colorado State >> IE 517

Course Hero has millions of student submitted documents similar to the one below including study guides, homework solutions, papers, exam answer keys and textbook solutions.

517, Math Assignment 2 Due Monday, September 29 Remember to provide full reasoning for all answers! 1. Show that every point of a set A X, where (X, d) is a metric space, is either an isolated point or a limit point. 2. Consider the convergence of sequences {xn } = {xn,1 , xn,2 , xn,3 , } to a point y = {y1 , y2 , y3 , } in 2. (a) Show that if xn y, then xn,i yi for i = 1, 2, 3, . But, construct an example that shows that xn,i yi for i = 1, 2, 3, does not imply that xn y. (b) Consider the subspace of 2, 2 H = x = {x1 , x2 , x3 , } : x and |xi | 1/i, i = 1, 2, 3, , called the Hilbert Cube. Show that a sequence {xn } = {xn,1 , xn,2 , xn,3 , } in H converges to a point y = {y1 , y2 , y3 , } if and only if xn,i converges to yi for i = 1, 2, 3, . 3. Let f : A R R have the property: for every x A there is an f (t) > if t (x , x + ) A. > 0 such that (We might say that f is locally bounded away from zero on A .) Show that if the set A is compact, there then is some c > 0 such that f (x) > c for all x A. 4. A function f : R R is locally increasing at a point x if there is a > 0 such that f (s) < f (x) < f (t) whenever x < s < x < t < x + . Show that a function that is locally increasing at every point in R must be increasing, i.e., f (x) < f (y) for all x < y. 5. Let (X, d) be a metric space. (a) Let {K1 , K2 , , Kn } be a set of compact subsets of X. Prove that n Ki is a compact subset i=1 of X. (b) Let {K } be a set of compact subsets of X. Prove that K is a compact subset of X. 6. Let (X, d) be a metric space with the discrete metric d (De nition 2.2.3). What subsets of X are compact? 7. Describe an open cover of the open unit square {(x, y) : 0 < x < 1, 0 < y < 1} that has no nite subcover. 8. Let {xn } be a sequence in a metric space (X, d) that converges to a limit x. Show the set {x, x1 , x2 , } is compact. 9. Show that the product of two compact metric spaces furnished with the product metric (Problem 6, Assignment 1) is a compact metric space. 1

Find millions of documents here - Study Guides, Homework Solutions, Papers, Exam Answer Keys and more. Course Hero has millions of course related materials that will enable you to learn better, faster and get an A in all your courses.
Below is a small sample set of documents: