Lebl-Basic-Analysis-I · Problem 2.3.16

Sequences and Series

Mathematics · limit-superior-limit-inferior-bolzano-weierstrass

Problem

Suppose {xn}n=1∞\{x_n\}_{n=1}^\infty is a bounded sequence, an:=sup⁡{xk:k≥n}a_n := \sup\{x_k : k \geq n\} as before. Suppose that for some ℓ∈N\ell \in \mathbb{N}, aℓ∉{xk:k≥ℓ}a_\ell \notin \{x_k : k \geq \ell\}. Then show that aj=aℓa_j = a_\ell for all j≥ℓj \geq \ell, and hence lim sup⁡n→∞xn=aℓ\displaystyle\limsup_{n\to\infty} x_n = a_\ell.

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