# converges to L

• Feb 17th 2010, 06:59 PM
redsoxnation
converges to L
Since I am not that great with finding subsequences I need help with writing this proof:

Let {x_n be a bounded sequence and suppose that every convergent subsequence of {x_n} converges to L. Prove that {x_n} converges to L.

I know that this statement does not guarantee that every subsequence converges. How would I go about the rest?
• Feb 17th 2010, 07:33 PM
Drexel28
Quote:

Originally Posted by redsoxnation
Since I am not that great with finding subsequences I need help with writing this proof:

Let {x_n be a bounded sequence and suppose that every convergent subsequence of {x_n} converges to L. Prove that {x_n} converges to L.

I know that this statement does not guarantee that every subsequence converges. How would I go about the rest?

Give us more than that.
• Mar 8th 2010, 08:08 PM
sfspitfire23
Quote:

Originally Posted by redsoxnation
Since I am not that great with finding subsequences I need help with writing this proof:

Let {x_n be a bounded sequence and suppose that every convergent subsequence of {x_n} converges to L. Prove that {x_n} converges to L.

I know that this statement does not guarantee that every subsequence converges. How would I go about the rest?

The Bolzano-Weierstrass theorem states that every bounded sequence will have a convergent subsequence. So, let $\epsilon>0$ and let there exist an N such that $|\{x_{n_k}\}-L|<\epsilon$ for $n_k>N$ and $\{x_{n_k}\}$ is a subsequence of $\{x_n\}$.....

no idea where to go from there...any input?