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I've been asked to do a proof showing that $d_1\left({x,y}\right)$ is a metric on $\mathbb{R}^k$, but is there any difference between this and $d_2\left({x,y}\right)$, for which I've already done a proof?

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No, $\sup$ and $\max$ are the same for finite sets. In fact, they are the same for infinite sets as well whenever both exist, however an infinite set of real numbers may fail to have a maximum even if it's bounded above (consider $\{1-\frac1n:n\in\mathbb N\}$).

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No, because the sets are finite.

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