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  1. MHB Craftsman
    Alexmahone's Avatar
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    #1
    Let $A$ and $B$ be sets of real numbers and write $$C=\{x+y:x\in A,y\in B\}.$$ Find a relation among $\sup A$, $\sup B$, and $\sup C$.

    My attempt:

    I'm assuming the answer is $\sup C=\sup A+\sup B$.

    $x\le \sup A\ \forall x\in A$
    $y\le \sup B\ \forall y\in B$
    $\implies x+y\le \sup A+\sup B$ $\forall x\in A,\ y\in B$

    So, $\sup A+\sup B$ is an upper bound for $C$.

    Suppose $\sup C\neq\sup A+\sup B$.

    $\implies \exists l<\sup A+\sup B$ such that $x+y\le l$ $\forall x\in A,\ y\in B$

    $\implies x\le l-y\ \forall x\in A,\ y\in B$

    So, $l-y$ is an upper bound of $A\ \forall y\in B$

    I feel that I'm on the right track but I don't know how to get a contradiction. Any suggestions?
    Last edited by Alexmahone; October 19th, 2016 at 10:05.

  2. MHB Master
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    #2
    Prove that $\sup A+\sup B-\varepsilon$ is not an upper bound of $C$ for any $\varepsilon>0$. For this find a $c\in C$ such that $c>\sup A+\sup B-\varepsilon$.

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