Show that a language with union is not regular by using pumping lemma Given the language $L:= { \{ c^{2k} w \ \vert \ k \ge 1, \ w \in \{a,b,c\}^* \ and \  \vert w\vert_a \ = \ \vert w\vert_b \} \  \cup \ \{ a,b \}^* }$
I'm really unsure how to even start because of the union.
I tried it with $w=a^nb^n$ but my correction said that it's
pumpable because it is in $\{ a,b \}^*$ which makes sense.
What would be a good word to start with?
I guess there are several cases i need to show?
If $L$ was regular, so would be $L \cap c^+\Sigma^* = \{c^{2k}w\mid k\geq 1, w\in\{a,b,c\}^* \text{ and }|w|_a = |w|_b\}$.
That means that if you prove that this language is not regular, then $L$ cannot be regular too (that way, we got rid of the union).
Now you can take it from here with pumping lemma, starting from $c^2a^nb^n$!