# Does a convergent subsequence require being bounded below and above?

As an extension to the title question:
If a sequence is bounded below (by say 0) but not above (infinity), does the Bolzano-Weierstrass theorem apply? (i.e. does it have a convergent subsequence?).

I think the answer is yes, since the subsequence can consist of those elements that tend towards zero but I'm not sure since I'm just getting familiar with the Bolzano-Weierstrass theorem.

Would like to kindly check my understanding. Thank you!

• Your title and questions do not seem to be the same – Henry Mar 12 '12 at 8:13
• @Henry: Apologies for this Henry. I've edited the question body to better reflect the title and the helpful answers provided by Marc and Aryabhata - it's really helped me a lot. – xlm Mar 12 '12 at 11:38

No, consider the sequence $x_n = n$.
If your question is "If a sequence is bounded below but not above, does it necessarily have a convergent subsequence?", then the answer is "no", as was indicated. However if your question is your title "Does a convergent subsequence require being bounded below and above?" then the answer is "yes" in either of the following two interpretations: (1) being convergent does require being bounded below and above (only finitely many terms can be at distance more than $1$ from the point of convergence), and also (2) you need the hypothesis of being bounded both above and below to ensure the existence of a convergent subsequence (as the given answers show). You probably meant question (2), but it would be good to choose your title so as to correspond to your actual question (and also in such a way that the answer is not the opposite to that of your actual question).