Basic Analysis I — Real Analysis · Problem 3.5.8

Continuous Functions — Jiri Lebl

Mathematics — Real Analysis · limits-at-infinity

Problem

Extend Lemma 3.5.5 as follows. Suppose S⊂RS \subset \mathbb{R} has a cluster point c∈Rc \in \mathbb{R}, c=∞c = \infty, or c=−∞c = -\infty. Let f:S→Rf : S \to \mathbb{R} be a function and suppose L=∞L = \infty or L=−∞L = -\infty. Show that lim⁡x→cf(x)=Lif and only iflim⁡n→∞f(xn)=L for all sequences {xn}n=1∞ such that lim⁡n→∞xn=c.\lim_{x \to c} f(x) = L \quad \text{if and only if} \quad \lim_{n \to \infty} f(x_n) = L \text{ for all sequences } \{x_n\}_{n=1}^\infty \text{ such that } \lim_{n \to \infty} x_n = c.

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