Measure, Integration & Real Analysis · Problem 2E.7

Integration — Sheldon Axler

Mathematics — Measure Theory · dini-theorem

Problem

Suppose

F

is a closed bounded subset of

\mathbf{R}

and

g_1, g_2, \ldots

is an increasing sequence of continuous real-valued functions on

F

(thus

g_1(x) \leq g_2(x) \leq \cdots

for all

x \in F

) such that

\sup\{g_1(x), g_2(x), \ldots\} < \infty

for each

x \in F

. Define a real-valued function

g

F

g(x) = \lim_{k \to \infty} g_k(x).

Prove that

g

is continuous on

F

if and only if

g_1, g_2, \ldots

converges uniformly on

F

g

. [The result above is called Dini's Theorem.]