Let (an) be a fixed, bounded sequence. For each n N+, set Sn = =...

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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 + ε} 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. 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 + ε} 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.

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