320 MAT Introduction to Analysis
Problem Set 6
due Friday, October 27
1. (a) Give an example of a sequence (xn ) which has subsequences (yn ), (zn ), and (wn ), such that lim yn = , lim zn = 0, and lim wn = 3.
n
n
n
Can the sequence (xn ) be convergent? (b) Give examples of two sequences (xn ) and (yn ) with
n
lim xn = lim yn = +
n
(and yn = 0 for all n), such that (i) limn (xn /yn ) = +; (ii) limn (xn /yn ) = 0; (iii) limn (xn /yn ) = 2; (iv) the sequence (xn /yn ) is bounded but divergent. 2. Suppose that xn 0 for all n N, and ((1)xn ) is a convergent sequence. Show that the sequence (xn ) is convergent. Whats its limit? Let 3. (xn ) be a bounded sequence, and let s = sup{xn : n N}. Also, suppose that s = xn for any n. Show that there is a subsequence of (xn ) which converges to s. What if xn = s for some n?
1 1 1 4. (a) Let xn = 1 + 2! + 3! + + n! . Show directly from the denition that (xn ) is a Cauchy sequence. 1 1 (b) Consider the sequence yn = 1 + 2 + + n . Show that (yn ) diver...