How to prove intersection between languages L1 (belongs to NP) and L2 (belongs to P) actually belongs to NP?
You can prove something more, intersection of two NP languages is also in NP.

Suppose you have polynomial time non-deterministic turing machines $M_1$ for $L_1$ and $M_2$ for $L_2$. We can modify $M_1$ as follows: Whenever any non-deterministic thread of $M_1$ accepts input string, say $w$, we run $M_2$ on $w$, if $M_2$ then accepts $w$, accept the input, otherwise reject.

The resulting machine is polynomial time non-deterministic.