This proof is a "translation" with my poor English of one which is given in Topologie, H. Queffelec (it seems that the proof is form A. Ancona). We assume that (if it's not the case we will take which is uniformly equivalent to ). Since is not compact we can find a sequence which has no accumulation point. Let where and .
We show that is equivalent to . Since we only have to show that is continuous. Let . Since the sequence has no accumulation points for all we can find such that for we have . Let . Since for we have we get and if then . This step shows that is a metric (the definition only gives that is a finite pseudometric).
Now we have to show that is a Cauchy sequence.
Let and with , and . We can find a such that .
First case : . We have .
Second case : . We have .