Heine-Cantor Theorem/Proof 1

Proof
Let $\epsilon \in \R_{>0}$ be a strictly positive real number.

For all $x \in A_1$, define:
 * $\map \Delta x = \set {\delta \in \R_{>0}: \forall y \in A_1: \map {d_1} {x, y} < 2 \delta \implies \map {d_2} {\map f x, \map f y} < \dfrac \epsilon 2}$

Define:
 * $\CC = \set {\map {B_{\delta} } x: x \in A_1, \, \delta \in \map \Delta x}$

where $\map {B_{\delta} } x$ denotes the open $\delta$-ball of $x$ in $M_1$.

From the definition of continuity, it follows that $\CC$ is a cover for $A_1$.

From Open Ball of Metric Space is Open Set, it therefore follows that $\CC$ is an open cover for $A_1$.

By the definition of a compact metric space, there exists a finite subcover $\set {\map {B_{\delta_1} } {x_1}, \map {B_{\delta_2} } {x_2}, \ldots, \map {B_{\delta_n} } {x_n} }$ of $\CC$ for $A_1$.

Define:
 * $\delta = \min \set {\delta_1, \delta_2, \ldots, \delta_n}$

Let $x, y \in A_1$ satisfy $\map {d_1} {x, y} < \delta$.

By the definition of a cover, there exists a $k \in \set{1, 2, \ldots, n}$ such that $\map {d_1} {x, x_k} < \delta_k$.

Then:

By the definition of $\map \Delta {x_k}$, it follows that:


 * $\map {d_2} {\map f x, \map f {x_k} } < \dfrac \epsilon 2$


 * $\map {d_2} {\map f y, \map f {x_k} } < \dfrac \epsilon 2$

Hence:

The result follows from the definition of uniform continuity.