This is a standard question about compact sets.

The way one does really depend on the tools you have.

We do not know that. I assume you know thatany nonempty set bounded set has a least upper bound.

Let such is covered by a finite collection of the .

Because for some then . So is not empty and bounded above by 1.

Let . So .

Suppose . Then some collection covers .

Say but .

But that means the same finite collection covers which contradicts the maximal nature of

Thus and the finite collection covers