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OK, so, given that $(X, \rho)$ is a metric space, endow $\mathbb{R}$ with the ordinary Euclidean metric $\varepsilon$ and $X \times X$ by $(\rho \times \rho) \left((x_1,y_1),(x_2,y_2)\right) = \sqrt{\rho(x_1,x_2)^2+\rho(y_1,y_2)^2}$.
Then, prove that
$\rho: X \times X \rightarrow \mathbb{R}_0^+, \quad (x,y)\mapsto \rho(x,y)$
is continuous.

I figure I probably have to use an $\epsilon-\delta$ proof, but it's resisted all of my efforts thus far.

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By triangle inequality you have:

$$|\rho (x_1,y_1)-\rho (x_2,y_2)| \leq |\rho (x_1,y_1)-\rho (x_2,y_1)| + |\rho (x_2,y_1)-\rho (x_2,y_2)| \; ,$$

hence, by reverse triangle inequality:

$$|\rho (x_1,y_1)-\rho (x_2,y_2)| \leq \rho (x_1,x_2) + \rho(y_1,y_2)\; ;$$

from the definition of $\rho\times \rho$ follows:

$$|\rho (x_1,y_1)-\rho (x_2,y_2)| \leq 2(\rho\times \rho) \big( (x_1,x_2) ,(y_1,y_2)\big) \; ,$$

therefore $\rho (x,y)$ is actually Lipschitz.

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