Identity Mapping between Metrics separated by Scale Factor is Continuous

Theorem
Let $M_1 = \struct {A, d_1}$ and $M_2 = \struct {A, d_2}$ be metric spaces on the same underlying set $A$.

Let $d_1$ and $d_2$ be such that:
 * $\forall x, y \in A: \map {d_2} {x, y} \le K \map {d_2} {x, y}$

Let $I_A: A \to A$ be the identity mapping on $A$.

Then $I_A$ is continuous from $M_1$ to $M_2$.

Proof
Let $\epsilon \in \R_{>0}$.

Let $a \in A$.

Set $\delta = \dfrac \epsilon K$.

Then:

Hence by definition $I_A$ is continuous at $a$.

As $a$ is arbitrary, it follows that this is true for all $a \in A$.

Thus $I_A$ is continuous on the whole of $M_1$.