Basic Analysis I — Real Analysis · Problem 3.5.7

Continuous Functions — Jiri Lebl

Mathematics — Real Analysis · limits-at-infinity

Problem

Let

\{x_n\}_{n=1}^\infty

be a sequence. Consider

S := \mathbb{N} \subset \mathbb{R}

, and

f : S \to \mathbb{R}

defined by

f(n) := x_n

. Show that the two notions of limit,

\lim_{n \to \infty} x_n \quad \text{and} \quad \lim_{x \to \infty} f(x)

are equivalent. That is, show that if one exists so does the other one, and in this case they are equal.