Basic Analysis I — Real Analysis · Problem 3.1.10

Continuous Functions — Jiri Lebl

Mathematics — Real Analysis · limits-of-functions

Problem

Suppose that

f : \mathbb{R} \to \mathbb{R}

be a function such that for every sequence

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

\mathbb{R}

, the sequence

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

converges. Prove that

f

is constant, that is,

f(x) = f(y)

for all

x, y \in \mathbb{R}