prove decidability and recognizability 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.
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&nbsp;$L_2$: every language is a subset of the decidable language&nbsp;$\Sigma^*$ 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. $L_1$&nbsp;is either decidable or recognizable; $L_2$&nbsp;is both. What do you know about the intersection of two decidable/recognizable languages?