Infinitely many elements?

Question

Given a set $A \subset \mathbb{R}$ bounded from above, $A \neq \emptyset$. If $A$ does not have a maximum, then does this imply that $A$ has infinitely many elements?

Any help would be appreciated!

Cheers

Answer 1

It is easy to show by induction that every finite nonempty totally ordered set has a maximal element.

Every subset $A\subset \mathbb R$ is totally ordered (since $\mathbb R$ is totally ordered), so if it does not have a maximal element it cannot be finite and nonempty by the above. So it is either infinite or empty.

Answer 2

Hint: Prove by induction that if $A(\subset\mathbb{R})$ has $n$ elements, then it has a maximum.