You are missing something, it should be .

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I will use the sequencial approach to limit because this approach is the nicest. Let be a sequence in converging to , i.e. . Then, we know (this is an easy theorem, but try to prove it) . Thus, is a sequence in approaching . Thus, this means . This implies then that . Q.E.D.

The theorem I used was let be a sequence of positive real numbers which is convergent to zero. Then . This theorem goes both ways.