Basic Analysis I — Real Analysis · Problem 6.1.12

Sequences of Functions — Jiri Lebl

Mathematics — Real Analysis · uniform-convergence-boundedness

Problem

Prove that if a sequence of functions

f_n : S \to \mathbb{R}

converge uniformly to a bounded function

f : S \to \mathbb{R}

, then there exists an

N

such that for all

n \geq N

, the

f_n

are bounded.