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!