Let be a metric space. Then

is a compatible

metric. Furthermore, is complete

implies is complete.

Proof.

Let . Let .

Now implies .

So .

Also, if , implies

, so

.

Now, assume that is complete. Then

let be a Cauchy sequence in . Let .

Then, .

Then, , so that is Cauchy in ,

hence in , and since ,

in . Thus, is complete.

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