True /False question about Compact sets and continuous functions

Question

I am trying some assignment questions and I am unable to think on how can I solve this problem.

Question: Let K be subset of $\mathbb{R^{n} }$ such that every real valued continuous function on K is bounded. Then is K compact?

I think this statement is true as if it were false then it would be impossible to give a counterexample as it has to be verified for every real valued function ( continuous) .

But I have no clue on how can I prove it.

Please give hint.

Does this answer your question? If every real-valued continuous function is bounded on $X$ (metric space), then $X$ is compact. — Brozovic, Jun 27 '20 at 19:25
You seem to be suggesting that to it is impossible to prove a "for all" statement, which is clearly absurd. — Lee Mosher, Jun 28 '20 at 18:32

score 1 · Accepted Answer · answered Jun 27 '20 at 19:26

Hint: $K$ is compact if and only if it is closed and bounded. You should devise a continuous function such that if $K$ is not bounded, the function is unbounded. Likewise, if $K$ is not closed, you can construct a continuous function on $K$ which is unbounded.

True /False question about Compact sets and continuous functions

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