# Help with a proof using the pumping lemma

I am confused with even starting the proof. I understand the pumping lemma:

Let A be a language over $$\Sigma$$. If A is regular, then there exists $$p > 0$$ (pumping length) such that $$∀s∈A$$, if $$|s|≥p$$, then there exists $$x,y,z∈ \Sigma^*$$ such that $$s = xyz$$ and:

1. $$|y|>0$$
2. $$|xy|≤p$$
3. $$∀i∈ℕ x(y^i)z∈A$$

What I am confused about is how does a computational trace end in a repeated state and have a length at most of length $$|Q|$$? How can I come about proving the statement using the pumping lemma?

• Doing the exercise does not require the use of the pumping lemma. If anything, the exercise is part of the proof of the pumping lemma, as $|Q|$ will be the pumping length, and the "length at most $|Q|$" condition of the exercise becomes condition 2 in the pumping lemma. – angryavian Feb 14 at 4:58
• This makes sense. My professor said that it would involve the use of the pumping lemma and I did not see it. I do see how it will be part of the proof though. Thanks! – Tom withers Feb 14 at 5:29