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Let $(A_n), (B_n)$ be two bounded sequences.

Show that there is a sequence of natural numbers $n_1 < n_2 <\cdots$ so that both the subsequences $(A_{n_k})$ and $(B_{n_k})$ converge.

My problem with solving this: Is it possible to say that assuming $A_n > B_n$ for all $n$ then we can make a subsequence of $n_1 < n_2 <\cdots$ from $B_n$ to $\infty$ and then from $A_n$ to $\infty$?

Thanks in advance!

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    First of all, as $(A_n)$ and $(B_n)$ don't relate in any way, it is sufficent, that each bounded sequence has a convergent subsequence. Therefore see http://en.wikipedia.org/wiki/Bolzano%E2%80%93Weierstrass_theorem2012-12-23
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    So what does that question mean? I'm puzzled...2012-12-23

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