# Prove that $\{(x_n,y_n)\}$ has a convergent subsequence.

The question is as follows:

Let $\{(x_n,y_n)\}$ be a sequence of points in $\mathbb R^2$ s.t. the sequences $\{x_n\}$ and $\{y_n\}$ are bounded. Prove that $\{(x_n,y_n)\}$ has a convergent subsequence.

Now, I am positive that I should tackle this problem using B-W theorem. Can anyone give an insight into this? Appreciate your help.

There exists a convergent subsequence $\;\{x_{n_k}\}\subset \{x_n\}\;$
Now look at $\;\{(x_{n_k}\,,\,y_{n_k})\}\;$ . This, again, is a bounded sequence, so again apply B-W to this sequence...on the second coordinate this time.
• In your third line, it may well be that the indexes $\;n_k\;$ for $\;x_n\;$ are not the same as the indexes for $\;y_n\;$ ...and still then I can't see any sound contradiction in your argument. What isn't clear in the above answer that cannot take you to the end of the proof in 2-3 lines more? – DonAntonio May 25 '14 at 18:23