Basic Analysis I — Real Analysis · Problem 6.2.7

Sequences of Functions — Jiri Lebl

Mathematics — Real Analysis · c1-norm-cauchy-sequence

Problem

For a continuously differentiable function

f : [a, b] \to \mathbb{R}

, define

\|f\|_{C^1} \mathrel{:=} \|f\|_{[a,b]} + \|f'\|_{[a,b]}.

Suppose

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

is a sequence of continuously differentiable functions such that for every

\epsilon > 0

, there exists an

M

such that for all

n, k \geq M

, we have

\|f_n - f_k\|_{C^1} < \epsilon.

Show that

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

converges uniformly to some continuously differentiable function

f : [a, b] \to \mathbb{R}