Lebl-Basic-Analysis-I · Problem 6.1.9

Sequences of Functions

Mathematics · uniform-convergence-monotone-functions

Problem

Let

f_n : [0,1] \to \mathbb{R}

be a sequence of increasing functions (that is,

f_n(x) \geq f_n(y)

whenever

x \geq y

). Suppose

f_n(0) = 0

and

\lim_{n \to \infty} f_n(1) = 0

. Show that

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

converges uniformly to

0