Let L be a language. w $\in$ L , and w could be broken in xyz.
Then if L is regular, there exists a pumping length p such that:
- |y| $\gt$ 0
- |xy| $\le$ p
- $\forall$ i $\ge$ 0, xy$^i$z $\in$ L
I understand that this lemma is based on pigeon hole principle and proof by contradiction.
For conditon (1), I understand that the string we are pumping should be $\ge$ 1.
But I don't understand the significance of condition (2). Why should the legth of xy be equal to or smaller than p . Will not it work otherwise?