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How can I prove $\sup(A+B)=\sup A+\sup B$ if $A+B= \lbrace a+b\mid a\in A, b\in B\rbrace $

here's a homework question I'm currently working on:

Let $A,B \subset \mathbb{R}$ non-empty sets bounded from above and from below. Show that $A+B$ is upper bounded and that $\sup(A+B)=\sup(A)+\sup(B)$

$A+B=\{a+b:a\in A, b \in B\}$

It was pretty easy to show to $A+B$ is upper-bounded by $\sup(A)+\sup(B)$, but I'm not quite sure how to prove that this is also the supremum. Any hints?


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marked as duplicate by lhf, Asaf Karagila, Martin Sleziak, Kannappan Sampath, Matt N. Mar 26 '12 at 11:58

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

Let $c$ some upper bound of $A + B$. Then for $a \in A$ $c-a$ is an upper bound of $B$, therefore ... –  martini Mar 26 '12 at 11:42
I think that I saw at least two of these questions posted on the site before. –  Asaf Karagila Mar 26 '12 at 11:49
Take $(a_n)_n \subset A$ to be the sequence such that $\lim_n a_n = \sup(A)$, similarly for $b_n$. What can you say about $(a_n+b_n)_n$ ? –  dtldarek Mar 26 '12 at 11:52

1 Answer 1

up vote 1 down vote accepted

Well $a+b\leq \sup(A)+sup(B)$ then $\sup(A+B) \leq \sup (A)+\sup(B)$ for the other inequality consider $a_{\epsilon}\in A$ such that $ a_{\epsilon}>\sup A- \epsilon /2$ and $b_{\epsilon}\in B$ such that $ b_{\epsilon}>\sup B- \epsilon /2$ then

$\sup(A+B)\geq a_{\epsilon}+b_\epsilon>\sup A +\sup B-\epsilon$ for any $\epsilon>0$ then

$\sup(A+B)\geq \sup A +\sup B$ $\blacksquare$

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