Show convergence of $x_n$ if $x_n$ is bounded and lim sup $x_n$ ⩽ $x_n$ for almost every n

Question

Im stucked on proofing that

If $x_n$ is bounded and lim sup $x_n$ ⩽ $x_n$ for almost every n, then $x_n$ converges.

Similary how would i show that this doesn't hold for every n?

Any help or hint how to approach a solution is much appreciated.

Thanks in advance!