To make sure: Pumping lemma for infinite regular languages only?

Question

So this is not about the pumping lemma and how it works, it\'s about a pre-condition.

Everywhere in the net you can read, that regular languages must pass the pumpin

Answer 1

There's a simplest way to show that some language is infinite. Assume that L is the language for some regular expression E, L(E).

Suppose L(E) is equivalent to {ab^nc | n ≥ 0}.

We know that E is in the form ab*c, and we know this language is probably regular(we can't prove something be regular), since the pumping lemma this conclusion is k = 0, put in another way, xz = ac, and every regular expression has an equivalent automaton.

The conclusion is simple, if the DFA has some state with transition to itself, the language is infinite.

     a   b   c

 q0  q1    
 q1     q1  q2
*q2         

q1 has transition to itself, thus L(E) is infinite.