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Im trying to proof Bolzano-Weierstraß in $\mathbb{R}^m$.

Could not find anything in the internet.

I suppose I have to consider the Bolzano-Weierstraß Theorem for the first dimension and use it for every row. But dont know how to "sum up" these results? With induction?

Any link/help?

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You can also do it all-at-once by a similar strategy: Split your compact set into a finite number of closed subsets (i.e., their union covers the original compact set). Then not all of these subsets can only contain a finite number of sequence elements. Pick one of the ones with infinitely many sequence elements and repeat the procedure starting from it.

Now you have only to enforce that the diameter of the subsequent subsets converges to zero (for instance, by splitting at the midpoint of the extend in each dimension) to get a nested sequence of subsets that converges to a point.

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