Lemma 1. Let be two metrics for the set X. If there exists a positive number such that (x,y) (x,y) for all x,y . Then the identity map X, d_2) \rightarrow (X, d_1)" alt="iX, d_2) \rightarrow (X, d_1)" /> is continuous.

Proof. Let and let be a positive number. If and x is a member of X for which , then . This shows that whenever . Thus X, d_2) \rightarrow (X, d_1)" alt="iX, d_2) \rightarrow (X, d_1)" /> is continuous.

I'll leave it to you to show that X, d_1) \rightarrow (X, d_2)" alt="i^{-1}X, d_1) \rightarrow (X, d_2)" /> is continuous.

Since X, d_2) \rightarrow (X, d_1)" alt="iX, d_2) \rightarrow (X, d_1)" /> is continuous, then for each -open set, is -open set. This argument applies for X, d_1) \rightarrow (X, d_2)" alt="i^{-1}X, d_1) \rightarrow (X, d_2)" /> as well. Thus, determines the same open sets in X.