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?

Have you made any progress? What do you know about compactness in a metric space? — Zach Stone, Jun 17 '15 at 02:07

score 2 · Answer 1 · edited Apr 13 '17 at 12:20

2

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.

edited Apr 13 '17 at 12:20

Community

1

answered Jun 17 '15 at 02:07

score 1 · Answer 2 · answered Jun 17 '15 at 02:10

1

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

answered Jun 17 '15 at 02:10

Irddo

2,317

Why is this function uniformly continuous?

2 Answers2