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Prove that $\mathrm{diam} (\overline{E})= \mathrm{diam} (E)$, where $E\subset (X,d)$, a metric space, and $\mathrm{diam}$ is the diameter of a set which is defined to be $\mathrm{diam}(E)$= $\sup \{d(p,q):\;p,q\in E\}$.

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