I've found a nice proof without 's...
Note that the sequence of general term , , is nondecreasing. This is because the terms of the sum are nonnegative ( for ), and each of them (i.e. ) increases with .
Plus, , hence converges to a finite limit.
However, and the sum converges to a finite limit as , hence must converge to a finite limit as well.
This limit has to be 0, for otherwise is the general term of a divergent series. Thus, .