Heine-Cantor: Continuity on Compact Interval Implies Uniform Continuity

If $f:[a,b]\to\mathbb{R}$ is continuous, then for every $\varepsilon>0$ there exists $\delta>0$ such that $|x-y|<\delta$ implies $|f(x)-f(y)|<\varepsilon$ for all $x,y\in[a,b]$.