pumping lemma for $L=\{a^n b^m c^k \mid n = m \vee m\neq k\}$

Question

Using pumping lemma, how can I prove that $L=\{a^n b^m c^k \mid n = m \vee m\neq k\}$ is not regular?.

If I choose $w= a^m b^m c^m$ and pump up with $i=2$, if have $a^m=1 b^m c^m$ but the string is still in the language. Any hint?

Answer 1

The easiest way to show that $L$ isn't regular is by noticing that $$ L \cap b^+c^+ = \{ b^m c^k : m,k \geq 1, m \neq k \}. $$ This should look familiar.