Call the space X, and the open subset Y. There are two things to prove here. First, that the identity map from Y with the metric D to Y with the metric D_1 is a homeomorphism. Second, that D_1 is a complete metric on Y.

Since

, it is clear that the identity map is continuous from (Y,D_1) to (Y,D). For the other direction, suppose that

in (Y,D). Then

, and it follows that

. That is (an outline of) the proof that the identity is a homeomorphism.

To show that (Y,D_1) is complete, let

be a Cauchy sequence in (Y,D_1). Then

is Cauchy in (Y,D) and so has a limit

. You have to show that

. For this, notice that

, and so

is a Cauchy sequence of real numbers. So it converges to a limit c. Therefore

. Hence

, which shows that

.