Lebl-Basic-Analysis-I · Problem 6.2.8

Sequences of Functions

Mathematics · l1-norm-uniform-convergence

Problem

Suppose

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

is a sequence of Riemann integrable functions on

[0,1]

that converges uniformly to

0

. Show that

\lim_{n\to\infty} \|f_n\|_{L^1} = 0,

where

\|f\|_{L^1} \mathrel{:=} \displaystyle\int_0^1 |f(x)|\, dx