prove decidability and recognizability
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?