Pumping Lemma Example 1

1. We don't know m, but assume there is one.

2. Choose a string w = aⁿbⁿ where n > m, so that any prefix of length m consists entirely of a's.

3. We don't know the decomposition of w into xyz, but since |xy| m, xy must consist entirely of a's. Moreover, y cannot be empty.

4. Choose i = 0. This has the effect of dropping |y| a's out of the string, without affecting the number of b's. The resultant string has fewer a's than b's, hence does not belong to L. Therefore L is not regular.