# Thread: Metric of a product

1. ## Metric of a product

Suppose that $\displaystyle (X, \rho ),(Y, \gamma )$ are metric spaces, define $\displaystyle ( X \times Y , \phi )$ with $\displaystyle \phi ((x_1,y_1),(x_2,y_2)) = \sqrt { ( \rho (x_1, x_2))^2+ ( \gamma (y_1,y_2))^2 }$

Show that $\displaystyle \phi ((x_1,y_1),(x_2,y_2)) + \phi ((x_2,y_2),(x_3,y_3)) \geq \phi ((x_1,y_1),(x_3,y_3))$

Proof so far.

I know that $\displaystyle \rho (x_1,x_2) + \rho (x_2,x_3) \geq \rho (x_1,x_3)$
and $\displaystyle \gamma (y_1,y_2) + \gamma (y_2,y_3) \geq \gamma (y_2,y_3)$

And I'm trying to show that $\displaystyle \sqrt { ( \rho (x_1, x_2))^2+ ( \gamma (y_1,y_2))^2 } + \sqrt { ( \rho (x_2, x_3))^2+ ( \gamma (y_2,y_3))^2 }$$\displaystyle \geq \sqrt { ( \rho (x_1, x_3))^2+ ( \gamma (y_1,y_3))^2 }$

How should I proceed? Thank you!

2. Hello,
$\displaystyle \sqrt {\rho (x_1, x_3)^2+\gamma (y_1,y_3)^2 } \leq \sqrt { \rho (x_1, x_2)^2+ \rho (x_2, x_3)^2 + \gamma (y_1,y_2)^2+ \gamma (y_2,y_3)^2 }.$
If $\displaystyle u$ and $\displaystyle v$ are non-negative real numbers then $\displaystyle \sqrt{u+v}\leqslant\sqrt{u}+\sqrt{v}$