Let (an) be a fixed, bounded sequence. For each n N+, set Sn = = sup{an, an+1, an+2,...} and tn = inf{an, an+1, an+2,...}. Also, set s = inf{sn: n N+} and t := sup{tn : n N+}. This s is called the limit superior ("lim sup") of (an) and denoted lim sup an or liman, while t is called the limit inferior ("lim inf") of (an) and denoted lim inf an or lim an. 1. Prove the following statements. c) [6 marks] For all e > 0, the set {n N+ : an < t + } is infinite. (Hint: Suppose not.) d) [6 marks] There is a subsequence of (an) that converges to t. (Hint: Apply Question 2(b) of Assignment 2.) If a sequence (an) is not bounded above we set lim supan = and if a sequence (an) is not bounded below we set lim inf an = -o. The results of Questions 1(d), 2(d), and 3 remain valid for unbounded sequences.