given closed subsets, A and B, of R^d with A bounded prove the equivalence of:(adsbygoogle = window.adsbygoogle || []).push({});

1) There exists a pair of sequences x_n in A and y_n in B such that |x_n - y_n| -> 0 as n -> infinity

2) A intersection B is non empty.

I have attempted this question but am a bit stuck on proving 1 implies 2 and not sure about my attempt at 2 implies 1.

For 2 implies 1:

Let p be a point of intersection between A and B.

Then let x_n be a sequence in A that tends to p as n -> infinity.

Let y_n be a sequence in B that tends to p as n -> infinity.

Then we have that ||x_n - y_n|| -> 0 as n ->infinity.

Thus we have shown a pair of sequences exists.

For 1 implies 2:

I'm actually clueless as to where to start. I've been told the Bolzano-Weierstrass theorem may help. I know this says that each bounded sequence in R^d has a convergent subsequence. I don't know how to use this though or really where to begin at all.

Any help would be appreciated.

Thanks in advance

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# Homework Help: Proving an equivalence concerning sequences

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