Basic Analysis I — Real Analysis · Problem 2.3.19

Sequences and Series — Jiri Lebl

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

Problem

Suppose {xn}n=1∞\{x_n\}_{n=1}^\infty is a bounded sequence, and ϵ>0\epsilon > 0 is given. Prove that there exists an MM such that for all k≥Mk \geq M, xk−(lim sup⁡n→∞xn)<ϵand(lim inf⁡n→∞xn)−xk<ϵ.x_k - \left(\limsup_{n\to\infty} x_n\right) < \epsilon \quad \text{and} \quad \left(\liminf_{n\to\infty} x_n\right) - x_k < \epsilon.

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