Convergence implies lim sup = lim inf Could someone please explain to me how the following can be proven? I get the intution but don't know how to write it rigorously.

Thank you.

Both $limsup$ and $liminf$ are limit points of a sequence; the largest and smallest limit points  respectively of a sequence, just as $lim$ is a limit point . Assuming your space is Hausdorff, or something else that guarantees that the limit is unique, then there is only one limit point, so we must have $liminf=limsup=lim$ 
