If $L$ is finite and $R$ is not regular, then $R\cup L$ is not regular

Question

Prove/Disprove: If $L$ is finite and $R$ is not regular, then $R\cup L$ is not regular.

I think that this one is true, but I am stuck:

Since $R$ is not regular, it is infinite, so $R \cup L$ is also infinite.

Since $R$ is non-regular, it must be that $R \cup L$ is not regular.

It just doesn't feel complete for me...

Any help will be amazing!

Thanks!

Answer 1

(weird choice of letter $R$ for a non-regular language).

Since $L$ is finite, so is $L\setminus R$ (hence regular).

Now, note that $R = (R\cup L) \setminus (L \setminus R)$. Can you find a contradiction if $R\cup L$ is regular?