Hey, can anyone help in this theorem.
If {Xn } is abounded sequence,then its limit exist as (n goes to infinity) if and only iff
Lim supXn = Lim infXn
n-->infinity n-->infinity
Use that $\displaystyle \liminf_{n\to\infty}x_n\le\lim_{n\to\infty}x_n\le\ limsup_{n\to\infty}x_n$ to prove the one part. Next consider instead of $\displaystyle limsup,\liminf$ think about it as $\displaystyle \sup\text{ }X,\inf\text{ }X$ where $\displaystyle X$ is the set of all subsequential limits of $\displaystyle \left\{x_n\right\}$. And use the fact that if every subsequential limit converges to the same value that the limit exists and equals that value.