Prove that $\sup{(A+B)} = \sup{A}+\sup{B}$ 
Let $A,B$ be subsets of $\mathbb{R}$. Prove that $\sup{(A+B)} = \sup{A}+\sup{B}$. 

I think in order to solve this we are going to have to use the mathematical definition of supremum. Maybe we can break this up into $4$ cases: $A$ is finite, $B$ infinite; etc. This would allow us to relate supremum to the maximal value of a set and make it easier to work with.

$A \leq \sup A$, $B \leq \sup B$, so $\sup A + \sup B \geq \sup(A+B)$
Also, $A + B \leq \sup (A+B)$, so $A \leq \sup(A+B) - B$, so $\sup A \leq \sup (A+B) -B$, then $\sup B \leq \sup (A+B) - \sup A$. Done
