The sequence, is a Cauchy sequence. Thus, for we have,

for any

The sequence, is a Cauchy sequence. Thus, for we have, for any

Consider the sequence, . For any we can find satisfy those two conditions on top. Let then,

and,

.

Addition yields,

Triangular inequality (Thanks to CaptainBlankfor teaching this move to me).

We have,

for .

Q.E.D.