I didn't get this on our first exam, and I'm trying to figure it out. I'm assuming Bolzano-Weierstrauss is used in this at some point, but I can't figure out how to formally prove the statement.

Claim: If $\displaystyle (a_{n})$ is a bounded sequence with the property that every convergent subsequence of $\displaystyle (a_{n})$ converges to the same limit a $\displaystyle \in \Re$, then $\displaystyle (a_{n})$ must converge.

Prove the Claim.