Assume $X_1$ and $X_2$ are two smooth projective curves and let $M_i$ be an ample line bundle on $X_i$, for $i=1,2$. Further, denote the natural projection map on the $i$-th factor by $$\pi_i:X_1\times X_2 \to X_i \,.$$
I need to show that the line bundle $\pi_1^* M_1\otimes\pi_2^* M_2$ over $X_1\times X_2$ is ample.
I know that, by Nakai's criterion, if $\pi_i$ were a finite morphism then the line bundle $\pi_i^* M_i$ would be ample. Nevertheless this is not the case here, as $\pi_i$ is not finite. I think $\pi_i^* M_i$ alone is not ample on the product space, but $\pi_1^* M_1\otimes\pi_2^* M_2$ actually is.
What could I do to see this? Any help is appreciated.