I am working on a pumping lemma question and trying to prove that the following is not context-free, but I can't finish the proof. The language is
$$L = \{ a^n b^m \mid n \leq m^2 \}$$
Assume Demon picks $p$ and I choose $s = a^{p^2}b^p$ and $s = uvwxy$ such that $p \geq |vwx|$.
The first case is $vwx = b^j$. If we choose $i = 0$ everything is ok and the relationship isn't right.
The second case is $vwx = a^j$. If we choose $i = 2$ every thing is ok.
I get stuck when $vwx = a^j b^k$. How can I finish the proof in this case?
Did I choose the right string? I repeated them for $s = a^{p}b^p$ but it didn't work.