Basic Analysis I — Real Analysis · Problem 2.3.12

Sequences and Series — Jiri Lebl

Mathematics — Real Analysis · limit-superior-limit-inferior

Problem

Let {xn}n=1∞\{x_n\}_{n=1}^\infty be a bounded sequence. a) Prove that there exists an ss such that for every r>sr > s, there exists an M∈NM \in \mathbb{N} such that for all n≥Mn \geq M, we have xn<rx_n < r. b) If ss is a number as in a), then prove lim sup⁡n→∞xn≤s\limsup_{n\to\infty} x_n \leq s. c) Show that if SS is the set of all ss as in a), then lim sup⁡n→∞xn=inf⁡S\limsup_{n\to\infty} x_n = \inf S.

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