# Closure properties between 2 languages of different types [duplicate]

Whenever said - The intersection between a Context Free Language and a Regular Language is always Context Free, what is the best logical way to confirm the statement?
I have this Chomsky hierarchy in mind that I refer whenever closure properties between Type-m and Type-n languages are asked but sometimes I come up with a wrong result.

How do you people logically solve it? What is the best simplest way?

You have to prove that the intersection of a context-free language and a regular language is always context-free. The simplest proof might go through push-down automata. Given a push-down automaton for a context-free language $L_1$, it is not too hard to augment it so that the state of a DFA for the regular language $L_2$ is maintained, and given this, you can construct a push-down automaton for $L_1\cap L_2$. Other proofs are also possible.