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I know that $\text{diam}(A)=\displaystyle\sup_{x,y\in A}d(x,y),$ but I don't see how to start the proof.

The thing I have is to let $\text{diam}(\overline A)=\displaystyle\sup_{x,y\in \overline A}d(x,y),$ so for $x,y\in \overline A,$ there exists sequences $x_n$ and $y_n$ such that $x_n\to x$ and $y_n\to y,$ but I don't know if I'm on the right track.