# finite automata – Using pumping lemma to show that there is always a smaller word for a regular language

I’m having trouble putting together a mathematical proof that uses the Pumping Lemma to show that $$exists$$n $$geqslant$$ 1 such that that for all strings w $$in$$ L such that |w| $$geqslant$$ n, there is another string z $$in$$ L such that |z| < n.

L is a regular language in this case.

I understand its saying that for any string in a regular language there can be a shorter version of it to an extent. But I’m not sure how to begin using the pumping lemma to show this. I have used the pumping lemma to show something is not regular, but I haven’t done a proof like this before…