Why is this function uniformly continuous?

Question

Assuming $X$ is a normed space. Why is a function $f:K\rightarrow\mathbb{R}$ uniformly continuous on a subspace $K\subset X$, if $K$ is sequentially compact and $f$ is continuous?

Answer 1

Because continuity on a compact is the same as uniform continuity.

Note that sequential compactness is the same as compactness.

Here's a link to a proof.

Answer 2

Hint: Prove that if $K, Y$ are metric spaces, with $K$ compact space, then every $f:K\to Y$ continuous is uniformly continuous.