There exists a variation of the pumping lemma with necessary and sufficient conditions for a language to be Regular.
According to that lemma:
A language $L$ is regular iff $\exists k$, $\forall x\in \Sigma^k$, $\exists u,v,w\in \Sigma^*$, $ x=uvw \cap |v|\ge 1$ such that:
$$\forall i \ge 0,\ \forall z\in \Sigma^*: uvwz\in L \iff uv^iwz\in L.$$
My question to you is: is there any way changing the for all $i \ge 0$ condition to for all $ 0\le i\le N$ for some $N$ - and the lemma will still be correct?
That $N$ may be constant, depend on the lemma's k, and so on.
I can't find an approach to prove it, any ideas?