What is the Convergence Criterion for a Bounded Sequence with a Common Limit?

[cragar]

Homework Statement

Assume $a_n$ is a bounded sequence with the property that every convergent sub sequence of $a_n$ converges to the same limit a. Show that
$a_n$ must converge to a.

The Attempt at a Solution

Could I do a proof by contradiction. And assume that $a_n$ does not converge
to a. but then this would imply that there would be a sub sequence that did not converge
to a and this is a contradiction because I could pick a sub sequence that converged to the same thing that $a_n$ did

 

[HallsofIvy]

Yes, that will work. Just fill in the details- why does the fact that $a_n$ does not converge to a imply that there exist a subsequence that does not converge to a? You will need to look at several cases- the sequence does not converge or it converges to some number other than a.

 

[cragar]

Could I say that eventually a sub sequence will have the same end behavior as
$a_n$ Or I could take 2 sub sequences that when put together would equal
$a_n$ Sub sequences aren't like subsets in the sense that a sub set could equal the set itself.