I'm trying to understand how to prove a language is regular or not regular, for example this language: $$L=\{a^nb^m:n,m\in\mathbb{N}\land n+m=5 \}$$ Is this language regular or not?
My solution
Using the pumping lemma, I can choose a string with a pumping length $p$ like: $w=a^{5-p}b^p$, then $x = a^j, y=a^l$ and $z=a^kb^p$ such that $j+l+k=5-p$, I will pump with $i=0$, so the string will be $xz=a^{j+k}b^p$, this is not regular because $j+k<p$.
Am I right about this ? Thanks for your help !