Suppose (a_{n}) is bounded, and that every convergent subsequence of (a_{n}) has limit L. Prove that $\displaystyle \lim_{n\rightarrow\infty}a_{n} = L$
Suppose the sequence does not converge to $\displaystyle L$.
What does that mean?
If it does converge to $\displaystyle L$ then if $\displaystyle \varepsilon > 0$ and $\displaystyle (L - \varepsilon ,L + \varepsilon )$ contains almost of the terms. So what is the negation of that?