[Heine–Cantor theorem] Prove by reduction to the absurd that if $K\subset\mathbb{R^m}$ is compact and $f: K\longrightarrow \mathbb{R}$ is continuous on $K$ then $f$ is uniformly continuous in $K$.
I know the proof by useing balls but I'm haveing a little trouble doing it by reduction to the absurd.
I'm trying to use the definition with sequences:
$f$ is not uniformly continuous in $K$ if there exists $\epsilon_0> 0$ such that for all $n \in\mathbb{N}$ there exist $x_n, y_n\in K$ such that $|x_n- y_n|<\frac{1}{n}\quad$ but $\quad|f (x_n)- f (y_n)| \geq \epsilon_0$.
Any suggestions about the process and what $x_n$ and $y_n$ could be would be great!