Lipschitz Equivalent Metrics are Topologically Equivalent

Theorem
Let $A$ be a set upon which there are two metrics imposed: $d_1$ and $d_2$.

Let $d_1$ and $d_2$ be Lipschitz equivalent.

Then $d_1$ and $d_2$ are topologically equivalent.

Proof
Consider the identity mapping:
 * $f: A \to A: \forall x \in A: f \left({x}\right) = x$

Then $f: \left({A, d_1}\right) \to \left({A, d_2}\right)$ can be considered as a Lipschitz equivalence.

The result then follows from Lipschitz Equivalent Metric Spaces are Homeomorphic.