Norm on a Cartesian product

Let $\displaystyle V_1, V_2$ be normed vector spaces with norms $\displaystyle ||\cdot||_1$ and $\displaystyle ||\cdot||_2$, respectively. Show that $\displaystyle (||v_1||_1^p + ||v_2||_2^p)^\frac{1}{p}, 1\leq p < \infty$ defines a norm on the space $\displaystyle V_1\times V_2$.

I've got the first three criteria down (positive definite, zero only when the vector is the zero vector, absolute homogeneity), however I'm having trouble seeing how to break this apart to show it satisfies the triangle inequality.

Suggestions on how to get started?