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Thread: Proving equivalences

  1. #1
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    Proving equivalences

    Let $\displaystyle X\ne\varnothing$ and let $\displaystyle c\in\mathbb R$ an upper bound for $\displaystyle X.$ Prove that the following sentences are equivalent:

    a) $\displaystyle c=\sup X.$

    b) For all $\displaystyle n>0$ exists an element $\displaystyle x\in X$ so that $\displaystyle c-\dfrac1n<x\le c.$

    First a) implies b): since $\displaystyle c=\sup X,$ then the number $\displaystyle c-\dfrac1n$ is not an upper bound of $\displaystyle X$ then there exists $\displaystyle x\in X$ so that $\displaystyle c-\dfrac1n<x\le c.$ Is that enough?

    I'm having problems to prove b) implies a), how to proceed?
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  2. #2
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    Suppose $\displaystyle c \neq \sup X$. Then there exists an upper bound c' < c for X...
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