Lebl-Basic-Analysis-I · Problem 3.1.13

Continuous Functions

Mathematics · limits-of-functions

Problem

Suppose

S \subset \mathbb{R}

and

c

is a cluster point of

S

. Suppose

f : S \to \mathbb{R}

is bounded. Show that there exists a sequence

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

with

x_n \in S \setminus \{c\}

and

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

such that

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

converges.