Bounded sequence, convergent subsequence

[kathrynag]

Homework Statement

Asssume (an) is a bounded sequence with the property that every convergent subsequence of (an) converges to the same limit a. Show that (an) must converge to a.

Homework Equations

The Attempt at a Solution

If the subsequence converges to a we have , we have:
$$\left|ank-a\right|$$=a.

 

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Choose $$n_k$$ large enough such that $$n_k \geq k \geq n_o$$. You should be to use this to show that your sequence converges.