0568 Q62 proof

[yoyostein]

Hi, could anyone give a proof of why Q62 II is true?

II. If every continuous real-valued function defined on K is bounded, then K is compact.
(K nonempty subset of R^n)

Thank you very much!

[blitzer6266]

First of all, in euclidean space, compact=closed and bounded

Suppose K is not compact, then either it is not closed or not bounded

Not bounded- consider the function ||x|| which will not be bounded
Not closed-there exists a limit point A such that x_n goes to A but A is not in K
Then you can look at the function 1/||x - A||. This will be continuous on K but unbounded since it goes to infinity as x goes to A

There are probably better proofs but this is at least the picture that should come to mind.

[yoyostein]

thanks!

this is crystal clear explanation