proof of Heine-Cantor theorem

We prove this theorem in the case when $X$ and $Y$ are metric spaces.

Suppose $f$ is not uniformly continuous. Then

In particular by letting $\delta =1/k$ we can construct two sequences $x_k$ and $y_k$ such that

Since $X$ is compact the two sequence have convergent subsequences i.e.

$$x_{k_j}\to \overline{x}\in X,y_{k_j}\to \overline{y}\in X.$$

Since $d(x_k,y_k)\to 0$ we have $\overline{x}=\overline{y}$. Being $f$ continuous we hence conclude $d(f(x_{k_j}),f(y_{k_j}))\to 0$ which is a contradiction being $d(f(x_k),f(y_k))\ge ϵ$.