The way one does really depend on the tools you have.
We do not know that. I assume you know that any 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