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I can't prove this theorem. Could you help me? Thanks for your helps.

$A$ and $B$ are subsets of an $n$-dimensional Euclidean space. Prove that, if $A$ is compact and $B$ is closed, there exist $u$ in $A$ and $v$ in $B$ such that $\|u-v\|=\inf\{\|a-b\| \mid a \in A\text{ and }b \in B\}$.

Bernard
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matholympian
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  • I think it's similar. This answer is for 1-dimensional euclidean space (real numbers), but i need an answer to n-dimensional. Anyway, maybe i can generalize this solution to my problem. Thank you very much. – matholympian Feb 15 '21 at 20:06

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You may assume that $B$ is compact by embedding $B$ inside a big enough cube (the intersection of this cube with $B$ is closed and bounded, so it's compact). You have to be a bit careful about how you choose this cube, but I'll let you think about that.

Then, $A\times B$ is compact so the function $(a,b) \mapsto \|a-b\|$ maps onto a compact set (image of compact is compact) and reaches its bounds (compact subsets of $\mathbb{R}$ are closed and bounded).

Sofía Marlasca Aparicio
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  • Maybe, in your answer, $A$ and $B$ are swapped, for $A$ is originally compact. So I think you wanted to say "you may assume that $B$ is compact" in the first line. Putting that aside, could you explain the part " the function $(a,b)\mapsto |a-b|$ maps onto a compact set and reached its bounds" in a little more detail ? I don't understand what the part means. – daㅤ Oct 15 '22 at 18:23
  • @daㅤedited my answer – Sofía Marlasca Aparicio Oct 17 '22 at 09:37