Proving the third limit theorem is a more elaborate task. At first we note that is bounded (cf. [5.5.1]), so that there is an with .
As we find an for such that
Furthermore there is an for so that
With these inequlities the following estimate holds for all :
Note that the last estimate is correct due to . (By the way: The strange addend 1 in the denominator only serves to cope with possibility .)