Is it possible to create a regular language from an non regular language? (details inside)
We have a language $L$ which is non-regular. We subtract a finite subset $S \subset L$ of words in $L$ to get the rest: $R = L \setminus S$.

Assume that $R$ is regular. Since regular languages are closed under union, and finite sets of words are trivially regular, we can construct a language $L' = R \cup S$ which is also regular.

Since $S \subset L$, we have $(L \setminus S) \cup S = L$, thus $L' = L$. But $L'$ is regular while $L$ is not - a contradiction. Thus our assumption that $R$ is regular was wrong.