Proof that the $\sup \left(a,b\right) = b$.

Your proof is correct!

However you can prove it directly via definition of least upper bound. As you have noted $b$ is an upper bound. Let's take any $c<b$ and show that $c$ is not a upper bound. Clearly any number between $\max(a, c) $ and $b$ lies in $(a, b) $ and is greater than $c$ and thus $c$ is not an upper bound. Thus $\sup(a, b) =b$.