MATH 131A-2 HOMEWORK 04 DUE ON OCT. 26 Exercise 1. Consider the sequence (sn ) defined by n X sin 2k 3 sn = k 2 k=1 for all positive integers n. Show that (sn ) is a Cauchy sequence. (Hint: use triangle inequality.) Exercise 2. Let (sn ) be a sequence. Assume that there is a sequence (ak ) such that for every k, there is a subsequence of (sn ) which converges to ak . Assume further that (ak ) converges to a. Show that there is a subsequence of (sn ) which converges to a. Exercise 3. Let (sn ) be a sequence. Let Pn sk s1 + + sn tn = k=1 = n n for all positive integer n. (1) Show that, for each positive number , and for each positive integer m, there is a positive integer N (, m) such that s1 + s2 + sm | 6 . | N (, m) Assume that (sn ) converges to s. (2) Show that, for each positive integer m, and for each integer n > m, we have s1 + + sm ms |sm+1 s| + + |sn s| |tn s| 6 | |+| |+ . n n n (3) Show that, for each \u000f > 0, there is an intger m and an integer N > m such that (i) |sn s| 6 3\u000f for each n > m, (ii) | s1 +s2n+sm | 6 3\u000f for each n > N (iii) | ms | 6 3\u000f for each n > N . n (4) Prove that (tn ) converges to s. (5) Now we do not assume that (sn ) converges to s any longer. We let sn = (1)n for all n. What can we say about the limits of (sn ) and (tn )? 1