Solution Verification: Prove or Disprove: If $L$ is an irregular language and $F$ is finite language, then $F\cap L^+$ is regular.

Question

Prove or Disprove:
If $L$ is an irregular language and $F$ is finite language, then $F\cap L^+$ is regular.

Note: $L^+=\bigcup_{i=1}^{\infty}L^i$.
I will be attempting to prove this statement.

First, I know that if a language is finite then it's regular, so I know for sure that $F$ is finite and regular. Now I don't really need to care if $L^+$ is finite or not and if it's irregular or not, because the intersection between a finite language $F$ and $L^+$ will always be finite, using the fact that $F\cap L^+ \subseteq F$.
And since it's a finite language, it will be regular.

Would appreciate any approvals and feedback, thanks in advance.

Answer 1

Looks good to me! More generally, take any finite language F and any language

L (whether L is regular, irregular, context free, ....),

their intersection must be finite, and hence their intersection is regular.