Assume that and for all  . Let . Choose such that . Next, for this choose  such that . Then, forimplies. Hence, .
Conversely, suppose that the limit of at does not exist. Then, there exists an such that for every there is some with

In particular, for each  there is some  with

Then for all , , but  . This is a contradiction. Hence the limit of at exists and is equal to .                                                                     Back