Proof That a Continuous Function on a Compact Metric Space is Uniformly Continuous
Ifis a continuous function,is a compact metric space andis a metric space thenis uniformly continuous.
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.