Thread: 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?

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.

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 and let there exist an N such that for and is a subsequence of .....

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