3: Let (an ) be a sequence of real numbers. Suppose that the subsequences (a2n ) and (a2n1) converge to the same number. Prove that (an ) converges. Problem 4: Suppose (an ) has two convergent subsequences, one that converges to a and one that converges to b, with a /= b. Prove that (an ) does not converge. Problem 5: Give an example of an unbounded sequence that has a convergent subsequence and one that does not have a convergent subsequence. Problem 6: Find a sequence (an ) with the following property: For every positive integer k, there exists a subsequence of (an ) that converges to k. Problem 7: Let r be a positive real number satisfying 0 < r < 1. For each positive integer n, let n . n+1 . r k= 1 r an = 1r k=0 Prove that (an ) converges. 1