Basic Analysis I — Real Analysis · Problem 6.2.15

Sequences of Functions — Jiri Lebl

Mathematics — Real Analysis · uniform-convergence-on-compact-subsets

Problem

We say that a sequence of functions

f_n : \mathbb{R} \to \mathbb{R}

converges uniformly on compact subsets if for every

k \in \mathbb{N}

, the sequence

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

converges uniformly on

[-k, k]

. (a)) Prove that if

f_n : \mathbb{R} \to \mathbb{R}

is a sequence of continuous functions converging uniformly on compact subsets, then the limit is continuous. (b)) Prove that if

f_n : \mathbb{R} \to \mathbb{R}

is a sequence of functions Riemann integrable on every closed and bounded interval

[a, b]

, and converging uniformly on compact subsets to an

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

, then for every interval

[a, b]

, we have

f \in \mathcal{R}([a,b])

, and

\int_a^b f = \lim_{n \to \infty} \int_a^b f_n