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I can't really wrap my head around $E$, or a Cauchy sequence in $E$. I need to take a Cauchy sequence in $E$ and show it's Cauchy in $(m,\left \| \cdot \right \|_\infty)$? I think I can show $(m,\left \| \cdot \right \|_\infty)$ is complete but I don't know how to use that info here.

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    Can you show that it is closed in $m$? Then use completeness of $m$. [for those not acquainted with the notation, $m$ is more commonly denoted by $\ell^\infty$]2012-02-16

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