# Supremum is the least upper bound

1. Nov 9, 2008

### sara_87

1. The problem statement, all variables and given/known data

Prove that the supremum is the least upper bound

2. Relevant equations

3. 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.

2. Nov 9, 2008

### HallsofIvy

Staff Emeritus
Please tell what your definitions of "supremum" and "least upper bound" are! As far as I know "supremum" is just another name for "least upper bound" and there is nothing to prove.