How to prove intersection between languages L1 (belongs to NP) and L2 (belongs to P) actually belongs to NP? I have to prove that if L &lt;=p L1 intersection L2, where L1 and L2 are described as above, L belongs to NP.

I thought about the definitions of P and NP and built a DTM D that decides L2 and a NTM N that decides L1. Then I tried to build a NTM NN that accepts when both D and N accept on some input w.

Now,did I really prove that intersection between L1 and L2 is in NP by constructing NTM NN? Thx in advance
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.