Basic Analysis I — Real Analysis · Problem 3.1.11

Continuous Functions — Jiri Lebl

Mathematics — Real Analysis · limits-of-functions

Problem

Prove the following stronger version of one direction of Lemma 3.1.7: Let

S \subset \mathbb{R}

c

be a cluster point of

S

, and

f : S \to \mathbb{R}

be a function. Suppose that for every sequence

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

S \setminus \{c\}

such that

\lim_{n \to \infty} x_n = c

the sequence

\bigl\{f(x_n)\bigr\}_{n=1}^\infty

is convergent. Then show that the limit of

f(x)

x \to c

exists.