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.

 

[╔(σ_σ)╝]

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.