Canonical Injection into Metric Space Product with P-Product Metric is Continuous/Proof 2

Proof
We want to show that:
 * $\forall c_1 \in A_1: \forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall x \in A_1: \map d {x, c} < \delta \implies \map {d_p} {\map {i_b} x, \map {i_b} c} < \epsilon$

and:
 * $\forall c_2 \in A_2: \forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall x \in A_1: \map {d'} {x, c} < \delta \implies \map {d_p} {\map {i_a} c, \map {i_a} x} < \epsilon$

Let $c_1$ and $c_2$ in $A_1$ and $A_2$ respectively be arbitrary.

Let $\epsilon \in \R_{>0}$ also be arbitrary.

Let $\delta = \epsilon$.

Let $x \in A_1$ such that $\map d {x, c_1} < \delta$.

We have:

We have that $c_1$ and $\epsilon$ are arbitrary.

Hence, by definition, $i_b$ is continuous in $M_1$.

Let $y \in A_2$ such that $\map d {y, c_2} < \delta$.

We have:

We have that $c_2$ and $\epsilon$ are arbitrary.

Hence, by definition, $i_a$ is continuous in $M_2$.