## Proof That a Continuous Function on a Compact Metric Space is Uniformly Continuous

Theorem

Ifis a continuous function,is a compact metric space andis a metric space thenis uniformly continuous.

Proof

TakeDefine an open cover ofas

Sinceis continuous, eachis open inandis an open cover of

is a compact metric space, so the above cover forhas a Lebesgue number

For eachthere existssuch thatandsuch that

From the definition of Lebesgue number,andis uniformly continuous.