Projection from Metric Space Product with Euclidean Metric is Continuous/Proof 2

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


 * $\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall z \in \AA: \map d {z, a} < \delta \implies \map {d_1} {\map {\pr_1} z, \map {\pr_1} a} < \epsilon$

and:
 * $\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall z \in \AA: \map d {z, a} < \delta \implies \map {d_2} {\map {\pr_2} z, \map {\pr_2} a} < \epsilon$

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 $\map d {x, a} < \delta$.

We have:

Then:

and similarly:

Hence:

and:

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

Hence the result by definition of ‎continuity.