proving Cartesian product of two metric space is a metric space

I saw somewhere that Cartesian product $X = X_1 \times X_2$ of two metric spaces $(X_1,d_1)$ and $(X_2,d_2)$ can be made into a metric space $(X,d)$ like following:

$d(x,y) = (d_1(x_1,y_1)^p + d_2(x_2,y_2)^p)^{1/p}$

How do I prove $(X,d)$ is metric space? More specifically, how to prove triangular inequality?

[Note: $p\ge1$, $x=(x_1,x_2)$ and $y= (y_1,y_2)$]

• the latter part $d_1 d_2$ is it $d_2(x_2, y_2)$? – Santosh Linkha Sep 27 '13 at 14:16
• yes sir,i corrected it – Raja Sekar Sep 27 '13 at 14:20
• "how to prove d(x,y) is metric space?" This is absurd, d(x,y) is a number, not a space. // Which axioms defining a metric space are a problem? – Did Sep 27 '13 at 14:23
• soory i corrected it – Raja Sekar Sep 27 '13 at 14:27
• Which axioms defining a metric space are a problem? – Did Sep 27 '13 at 14:29

Hint: Suppose $X_1 = X_2 = \mathbb{R}$. You know how to make a metric on $\mathbb{R} \times \mathbb{R} = \mathbb{R}^2$ out of the metric on $\mathbb{R}$ in a way that looks like the one above (think: Pythagorean theorem). Try and generalize this.