# How to show that $\sup(A + B) = \sup\{x + y : x \in A, y \in B\} = \sup A + \sup B$ for bounded subsets $A, B$ of $\mathbb{R}$

Let $$A$$ and $$B$$ be two non-empty bounded subsets of $$\mathbb{R}$$. Write $$A + B = \{x + y : x \in A, y \in B\}$$. Show that $$\sup(A + B) = \sup A + \sup B$$.

This is a problem from an assignment from my analysis course. the two definitions I know of supremum is that it is the least upper bound and also for all $$\epsilon >0 \sup(S)-\epsilon < a$$ for some $$a$$ belonging to the set $$S$$.

I am unable to prove this using the above definitions (though the second one is actually a theorem that can be proven from the previous definition). I have already tried writing the definition of supremum and saying that for all $$\epsilon >0$$, there exists $$a+b$$ belonging to $$A+B$$ such that $$\sup(A+B) But I am unable to proceed further.

Can anyone please provide a right approach to solve this problem and what lemmas must be proven.

• What have you attempted? – ncmathsadist Sep 23 '18 at 19:19
• Can you at least see why $\sup A+\sup B$ is an upper bound of $A+B$? Given that, can you see how to proceed? – Crosby Sep 23 '18 at 19:22
• I have edited the question to show how much I have actually tried. To be honest I am beginner to analysis and have not done a lot of progress on this question. – NKB Sep 23 '18 at 19:23
• @Crosby Yep I see it's an UB. – NKB Sep 23 '18 at 19:23
• @NKB Okay, now can you show it is the least upper bound? What do you know about $\sup A$ and $\sup B$, by definition. – Crosby Sep 23 '18 at 19:24

You wrote in the comments that you see why $$supA+supB$$ is an upper bound of the set $$A+B$$. Now you just have to prove it is the least upper bound. Alright, so fix $$\epsilon>0$$. Then you know there exists $$a\in A$$ such that $$a>supA-\frac{\epsilon}{2}$$ and there exists $$b\in B$$ such that $$b>supB-\frac{\epsilon}{2}$$. So then $$a+b\in A+B$$ and $$a+b>supA-\frac{\epsilon}{2}+supB-\frac{\epsilon}{2}=supA+supB-\epsilon$$. Hence anything smaller than $$supA+supB$$ is not an upper bound of $$A+B$$.