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

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:
{{begin-eqn}}
{{eqn | l = \map {d_p} {z, a}
      | r = \map {d_p} {x, y} := \paren {\paren {\map d {x_1, y_1} }^p + \paren {\map {d'} {x_2, y_2} }^p}^{1/p}
      | c = Definition of $d$
}}
{{eqn | l = \map d {\map {\pr_1} z, \map {\pr_1} a}
      | r = \map d {x_1, x_0}
      | c = {{Defof|First Projection}}
}}
{{eqn | l = \map {d'} {\map {\pr_2} z, \map {\pr_2} a}
      | r = \map {d'} {y_1, y_0}
      | c = {{Defof|Second Projection}}
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \paren {\map d {x_1, x_0} }^p + \paren {\map {d'} {y_1, y_0} }^p
      | o = \ge
      | r = \paren {\map d {x_1, x_0} }^p
      | c = 
}}
{{eqn | n = 1
      | ll= \leadsto
      | l = \paren {\paren {\map d {x_1, y_1} }^p + \paren {\map {d'} {x_2, y_2} }^p}^{1/p}
      | o = \ge
      | r = \map d {x_1, x_0}
      | c = 
}}
{{end-eqn}}
and similarly:
{{begin-eqn}}
{{eqn | l = \paren {\map d {x_1, x_0} }^p + \paren {\map {d'} {y_1, y_0} }^p
      | o = \ge
      | r = \paren {\map {d'} {y_1, y_0} }^2
      | c = 
}}
{{eqn | n = 2
      | ll= \leadsto
      | l = \paren {\paren {\map d {x_1, y_1} }^p + \paren {\map {d'} {x_2, y_2} }^p}^{1/p}
      | o = \ge
      | r = \map {d'} {y_1, y_0}
      | c = 
}}
{{end-eqn}}
Hence:
{{begin-eqn}}
{{eqn | l = \map {d_p} {z, a}
      | o = <
      | r = \delta
      | c = 
}}
{{eqn | ll= \leadsto
      | l = \paren {\paren {\map d {x_1, y_1} }^p + \paren {\map {d'} {x_2, y_2} }^p}^{1/p}
      | o = <
      | r = \delta
      | c = Definition of $d_p$
}}
{{eqn | ll= \leadsto
      | l = \map d {x_1, x_0}
      | o = <
      | r = \delta
      | c = from $(1)$
}}
{{eqn | ll= \leadsto
      | l = \map d {\map {\pr_1} z, \map {\pr_1} a}
      | o = <
      | r = \epsilon
      | c = as $\epsilon = \delta$
}}
{{end-eqn}}
and:
{{begin-eqn}}
{{eqn | l = \map {d_p} {z, a}
      | o = <
      | r = \delta
      | c = 
}}
{{eqn | ll= \leadsto
      | l = \paren {\paren {\map d {x_1, y_1} }^p + \paren {\map {d'} {x_2, y_2} }^p}^{1/p}
      | o = <
      | r = \delta
      | c = Definition of $d_p$
}}
{{eqn | ll= \leadsto
      | l = \map {d'} {y_1, y_0}
      | o = <
      | r = \delta
      | c = from $(2)$
}}
{{eqn | ll= \leadsto
      | l = \map {d'} {\map {\pr_2} z, \map {\pr_2} a}
      | o = <
      | r = \epsilon
      | c = as $\epsilon = \delta$
}}
{{end-eqn}}
We have that $a$ and $\epsilon$ are arbitrary.
Hence the result by definition of continuity.
{{qed}}
Category:Projections
Category:P-Product Metrics
Category:Continuous Mappings on Metric Spaces
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