Problem 3: Let (an ) be a sequence of real numbers. Suppose that the subsequences (a2n )and (a2n−1) converge to the same number. Prove that (an ) converges.•Problem 4: Suppose (an ) has two convergent subsequences, one that converges to a andone 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 subsequenceand 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,letn.− n+1.r k= 1 ran =1−rk=0Prove that (an ) converges.•1