Pumping-Lemma regular languages: Consider multiple cases?
Of course, you have to consider all possible cases. Otherwise you do not prove that "there exists a factorization" is false. So you actually have to consider all $ab^k$ with $k &lt; n$. Here you get some prefix $ab\cdots bab\cdots b$ and therefore the words are outside tha language. Or you chose a different word $z$ if $n=0$ is allowed, which depends on your definition of the natural numbers - then you only have one case.

For the case you treat, probably stating that $uv^2w = ab^{n+k}c^n$ would be more complete than saying "obviously."