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.
, 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
, which shows that