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.
prove decidability and recognizability
formal languages;computability;regular languages;closure properties
The method you propose in your answer is mostly correct. $L_2$ is, indeed, decidable because it's regular. But you can't argue that $L_1\cap L_2$ is decidale because it is a subset of the decidable language $L_2$: every language is a subset of the decidable language $\Sigma^*$ but that doesn't mean that every langauge is decidable.Hint. You chose the tag closure-properties, which shows you're on the right lines. $L_1$ is either decidable or recognizable; $L_2$ is both. What do you know about the intersection of two decidable/recognizable languages?