- #1

- 361

- 48

## Main Question or Discussion Point

I'm seeking a neater proof for the following:

Let E [itex]\subseteq[/itex] R. Let every continuous real-valued function on E be bounded. Show that E is compact.

I tried to argue based on Heine-Borel theorem as follows:

E cannot be unbounded because if it is the case, define f(x)=x on E and f(x) is continuous but unbounded.

E cannot be not closed, because E=(0,1] is not closed and we can define f(x)=1/x which is continuous but still unbounded.

Thus, E has to be both bounded and closed and hence compact.

I think this argument is not very solid and I'd appreciate any hints.

Let E [itex]\subseteq[/itex] R. Let every continuous real-valued function on E be bounded. Show that E is compact.

I tried to argue based on Heine-Borel theorem as follows:

E cannot be unbounded because if it is the case, define f(x)=x on E and f(x) is continuous but unbounded.

E cannot be not closed, because E=(0,1] is not closed and we can define f(x)=1/x which is continuous but still unbounded.

Thus, E has to be both bounded and closed and hence compact.

I think this argument is not very solid and I'd appreciate any hints.