Earlier on in the book it showed that to prove $a = b$ it is often best to show that $a \leq b$ and that $b \leq a$. This is the way I want to go about the proof. I am sure there is an easier way but I am new to this and I am trying to get a basic grasp on everything before I start class in the fall. Forgive my terrible MathJax.
Question If $A + B := \{a + b | a \in A \land b \in B\}$ Where $A \space and \space B$ are non-empty, bounded subsets of $\mathbb{R}$ then $\sup(A+B) = \sup(A)+\sup(B)$
Since both A and B are non-empty bounded subsets of $\mathbb{R}$ if follows that both A and B have a finite Supremum. Thus $\sup(A)$ and $\sup(B)$ exist. Obviously $\sup(A) \geq a \space \space \forall a \in A$ and $\sup(B) \geq b \space \space \forall \space b \in B$. So $\sup(A) + \sup(B) \geq A + B$ Which means $\sup(A) + \sup(B)$ is an upper bound of $A + B$ and therefore it follows that $\sup(A) + \sup(B) \geq \sup(A + B)$
Likewise since A and B are non-empty bounded subsets of $\mathbb{R}$ It follows that the set $A + B := \{a + b| a \in A \land b \in B\}$ is also a finite bounded subset of $\mathbb{R}$ Therefore the set $A + B$ has a finite \supremum so $\sup(A + B)$ exists. Since $\sup(A) + \sup(B) \in A + B$. It then directly follows that $\sup(A+B)$ is an upperbound of $\sup(A) + \sup(B)$ and then it directly follows that $\sup(A+B) \geq \sup(A) + \sup(B)$
Which completes the proof that $\sup(A) + \sup(B) = \sup(A + B)$.