Projection from Metric Space Product with P-Product Metric is Continuous

Theorem
Let $M_1 = \struct {A_1, d}$ and $M_2 = \struct {A_2, d'}$ be metric spaces.

Let $\AA := A_1 \times A_2$ be the cartesian product of $A_1$ and $A_2$.

Let $\MM = \struct {\AA, d_p}$ denote the metric space on $\AA$ where $d_p: \AA \to \R$ is the $p$-product metric on $\AA$:


 * $\map {d_p} {x, y} := \paren {\paren {\map d {x_1, y_1} }^p + \paren {\map {d'} {x_2, y_2} }^p}^{1/p}$

where $x = \tuple {x_1, x_2}, y = \tuple {y_1, y_2} \in \AA$.

Let $\pr_1: \MM \to M_1$ and $\pr_2: \MM \to M_2$ denote the first projection and second projection respectively on $\MM$.

Then $\pr_1$ and $\pr_2$ are both ‎continuous on $\MM$.

Proof
We want to show that, for all $a \in \AA$:


 * $\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall z \in \AA: \map {d_p} {z, a} < \delta \implies \map d {\map {\pr_1} z, \map {\pr_1} a} < \epsilon$

and:
 * $\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall z \in \AA: \map {d_p} {z, a} < \delta \implies \map {d'} {\map {\pr_2} z, \map {\pr_2} a} < \epsilon$

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

Let $a = \tuple {x_0, y_0} \in \AA$ also be arbitrary.

Let $\delta = \epsilon$.

Let $z = \tuple {x_1, y_1} \in \AA$ such that $\map {d_p} {x, a} < \delta$.

We have:

Then:

and similarly:

Hence:

and:

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

Hence the result by definition of ‎continuity.