I'm learning the proof of the Pumping Lemma for regular languages. The proof is carried out using an arbitrary string having length of at least the number of states in the DFA. As such:
The language L is regular, so there exists a DFA M such that L = L(M). Say M has p states, {q1, . . . , qp}. We are also given input string s ∈ L with s = s1s2 · · · sn (n = |s| ≥ p).
How do we know that the DFA would accept such a string? For example, if we consider this DFA:
https://i.stack.imgur.com/4R0zt.png
This DFA has 4 states but doesn't accept any string that has length of more than two. So I'm confused about this aspect of the proof.