Heine-Cantor Theorem/Proof 1

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Theorem

Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ be metric spaces.

Let $M_1$ be compact.

Let $f: A_1 \to A_2$ be a continuous mapping.


Then $f$ is uniformly continuous.


Proof

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


For all $x \in A_1$, define:

$\Delta \left({x}\right) = \left\{{\delta \in \R_{>0}: \forall y \in A_1: d_1 \left({x, y}\right) < 2 \delta \implies d_2 \left({f \left({x}\right), f \left({y}\right)}\right) < \dfrac \epsilon 2}\right\}$

Define:

$\mathcal C = \left\{{B_{\delta} \left({x}\right): x \in A_1, \, \delta \in \Delta \left({x}\right)}\right\}$

where $B_{\delta} \left({x}\right)$ denotes the open $\delta$-ball of $x$ in $M_1$.

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

From Open Ball is Open Set, it therefore follows that $\mathcal C$ is an open cover for $A_1$.


By the definition of a compact metric space, there exists a finite subcover $\left\{{B_{\delta_1} \left({x_1}\right), B_{\delta_2} \left({x_2}\right), \ldots, B_{\delta_n} \left({x_n}\right)}\right\}$ of $\mathcal C$ for $A_1$.

Define:

$\delta = \min \left\{{\delta_1, \delta_2, \ldots, \delta_n}\right\}$


Let $x, y \in A_1$ satisfy $d_1 \left({x, y}\right) < \delta$.

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

Then:

\(\displaystyle d_1 \left({y, x_k}\right)\) \(\le\) \(\displaystyle d_1 \left({y, x}\right) + d_1 \left({x, x_k}\right)\) Triangle Inequality
\(\displaystyle \) \(<\) \(\displaystyle \delta + \delta_k\) Metric Space Axiom $\left({M3}\right)$
\(\displaystyle \) \(\le\) \(\displaystyle 2 \delta_k\)


By the definition of $\Delta \left({x_k}\right)$, it follows that:

$d_2 \left({f \left({x}\right), f \left({x_k}\right)}\right) < \dfrac \epsilon 2$
$d_2 \left({f \left({y}\right), f \left({x_k}\right)}\right) < \dfrac \epsilon 2$

Hence:

\(\displaystyle d_2 \left({f \left({x}\right), f \left({y}\right)}\right)\) \(\le\) \(\displaystyle d_2 \left({f \left({x}\right), f \left({x_k}\right)}\right) + d_2 \left({f \left({x_k}\right), f \left({y}\right)}\right)\) Triangle Inequality
\(\displaystyle \) \(<\) \(\displaystyle \frac \epsilon 2 + \frac \epsilon 2\) Metric Space Axiom $\left({M3}\right)$
\(\displaystyle \) \(=\) \(\displaystyle \epsilon\)

The result follows from the definition of uniform continuity.

$\blacksquare$


Source of Name

This entry was named for Heinrich Eduard Heine and Georg Cantor.


Sources