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I am to assume that $K$ is a compact metric space. I must prove that there are two points $x,y$ contained in $K$ such that $d(x,y)=\text{diam}(K)$.

Recall $\text{diam}(K)= \sup \{ d(x,y) \mid x,y \in K \}$.

I'm having trouble figuring out exactly what I need to do to get started on this proof and what it is I need to prove along the way. I know the definitions, but I'm having trouble applying them to this problem.


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The metric $d : (x,y) \mapsto d(x,y)$ is continuous, so by compactness $d$ has a maximum on $K \times K$.

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