I want to prove that for any language $L_1$ described by a Turing machine and any regular language $L_2$, $L_1 \cap L_2$ is described by a Turing machine that its recognizability and decidability is same as $L_1$.
I thought so far that I can describe $L_2$ with a Turing machine. So I can next find the intersection Turing machine of both.
I do not know what to do about the last part of the problem. Maybe I can say that $L_2$ is decidable (because it is regular) and $L_1 \cap L_2$ is a subset of that. So it is also decidable.
I am not sure whether my approach is correct or not. Please help me.