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Possible Duplicate:
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?

Thanks!

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marked as duplicate by lhf, Asaf Karagila, Martin Sleziak, user21436, Rudy the Reindeer Mar 26 '12 at 11:58

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    $\begingroup$ Let $c$ some upper bound of $A + B$. Then for $a \in A$ $c-a$ is an upper bound of $B$, therefore ... $\endgroup$ – martini Mar 26 '12 at 11:42
  • $\begingroup$ I think that I saw at least two of these questions posted on the site before. $\endgroup$ – Asaf Karagila Mar 26 '12 at 11:49
  • $\begingroup$ 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 $(a_n+b_n)_n$ ? $\endgroup$ – dtldarek Mar 26 '12 at 11:52
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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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  • $\begingroup$ Isnt't there a dot missing prior to "for the other..."? $\endgroup$ – Fabian Schn. Nov 15 '17 at 11:33

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