Lipschitz Equivalent Metrics are Topologically Equivalent/Proof 3

Proof
By definition of Lipschitz equivalence:

$\exists K_1, K_2 \in \R_{>0}$ such that:
 * $(1): \quad \forall x, y \in A: \map {d_2} {x, y} \le K_1 \map {d_1} {x, y}$
 * $(2): \quad \forall x, y \in A: \map {d_1} {x, y} \le K_2 \map {d_2} {x, y}$

By Identity Mapping between Metrics separated by Scale Factor is Continuous:
 * the identity mapping from $M_1$ to $M_2$ is continuous
 * the identity mapping from $M_2$ to $M_1$ is continuous.

Hence the result by definition of topological equivalence.