Proof of the Heine–Cantor theorem by a finite cover of balls

Proof

We use an open cover by small balls and take the smallest radius of a finite subcover.

1. Write $d_X$ and $d_Y$ for the metrics of $X$ and $Y$. Fix a real $\varepsilon > 0$. We must find $\delta > 0$ with $d_Y(f(x), f(y)) < \varepsilon$ whenever $d_X(x, y) < \delta$.
2. If $X$ is empty, every $\delta > 0$ works. From now on let $X$ be nonempty.
3. Let $p \in X$. The map $f$ is continuous, so preimages of open sets are open. By continuity via open sets, there is $\delta_p > 0$ with $d_Y(f(x), f(p)) < \varepsilon / 2$ for all $x$ with $d_X(x, p) < \delta_p$.
4. The open balls $B(p, \delta_p / 2)$, for $p \in X$, are open. Each $p$ lies in its own ball, so they form an open cover of $X$.
5. Since $X$ is compact, finitely many of them cover $X$. Call their centres $p_1, \ldots, p_n$, with $n \ge 1$, and write $\delta_k = \delta_{p_k}$.
6. Let $\delta = \min(\delta_1 / 2, \ldots, \delta_n / 2)$. This is a minimum of finitely many positive reals, so $\delta > 0$.
7. Let $x, y \in X$ with $d_X(x, y) < \delta$. By step 5, $x \in B(p_k, \delta_k / 2)$ for some $k$. So $d_X(x, p_k) < \delta_k / 2 < \delta_k$.
8. By the triangle inequality, $d_X(y, p_k) \le d_X(y, x) + d_X(x, p_k) < \delta + \delta_k / 2 \le \delta_k$.
9. By steps 7 and 8 and the choice of $\delta_k$ in step 3, $d_Y(f(x), f(p_k)) < \varepsilon / 2$ and $d_Y(f(y), f(p_k)) < \varepsilon / 2$.
10. By the triangle inequality in $Y$, $d_Y(f(x), f(y)) < \varepsilon / 2 + \varepsilon / 2 = \varepsilon$.
11. The number $\delta$ depends only on $\varepsilon$. So condition 1 of uniform continuity holds, and $f$ is uniformly continuous.