1. ## 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?

2. 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.

3. 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 $\displaystyle \epsilon>0$ and let there exist an N such that $\displaystyle |\{x_{n_k}\}-L|<\epsilon$ for $\displaystyle n_k>N$ and $\displaystyle \{x_{n_k}\}$ is a subsequence of $\displaystyle \{x_n\}$.....

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