Metric is Continuous

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Theorem

Let $M = \left({A, d}\right)$ be a metric space.

Let $\tau_A$ be the topology on $A$ induced by $d$.

Let $\left({A \times A, \tau}\right)$ be the topological product of $\left({A, \tau_A}\right)$ and itself.


Then the metric $d: A \times A \to \R$ is a continuous mapping.


Proof

Let $d_{\infty}: \left({A \times A}\right) \times \left({A \times A}\right) \to \R$ be the metric on $A \times A$ defined by:

$d_{\infty} \left({\left({x, y}\right), \left({x', y'}\right)}\right) = \max \left\{{d \left({x, x'}\right), d \left({y, y'}\right)}\right\}$

By Product Space Metric Induces Product Topology, $\tau$ is the topology on $A \times A$ induced by $d_{\infty}$.


Let $\epsilon \in \R_{>0}$ be a strictly positive real number.

Let $\left({x_0, y_0}\right) \in A \times A$.

Suppose that $\left({x, y}\right) \in A \times A$ and $d_{\infty} \left({\left({x, y}\right), \left({x_0, y_0}\right)}\right) < \dfrac 1 2 \epsilon$.

Then:

\(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle \left\vert{d \left({x, y}\right) - d \left({x_0, y_0}\right)}\right\vert\) \(\le\) \(\displaystyle \) \(\displaystyle \left\vert{d \left({x, y}\right) - d \left({x_0, y}\right)}\right\vert + \left\vert{d \left({x_0, y}\right) - d \left({x_0, y_0}\right)}\right\vert\) \(\displaystyle \) \(\displaystyle \)          by the triangle inequality          
\(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\le\) \(\displaystyle \) \(\displaystyle d \left({x, x_0}\right) + d \left({y, y_0}\right)\) \(\displaystyle \) \(\displaystyle \)          by the reverse triangle inequality          
\(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(<\) \(\displaystyle \) \(\displaystyle \epsilon\) \(\displaystyle \) \(\displaystyle \)          by the definition of $d_{\infty}$          

The result follows from the definition of a continuous mapping.

$\blacksquare$