# Supremum is the least upper bound

## Homework Statement

Prove that the supremum is the least upper bound

## The Attempt at a Solution

Proof: let x be an upper bound of a set S then x>=supS (by definition). If there exists an upper bound y and y<=SupS then y is not an upper bound (contradiction) therefore every upper bound is greater than SupS so SupS is the least upper bound.

Is that proof correct?

Thank you.