How to prove language $L=\{a^{i}b^{j} : i \leq j^{2}\}$ is not CFL using Pumping lemma?
Suppose towards a contradiction that  $L$ is context-free and let $c$ be the pumping length of $L$.
Consider the word $a^{c^2} b^c$ which, by the pumping lemma, can be written as $uvwxy$ with $1 \le |vx| \le c$.

Notice that:


$vx$ cannot contain only $b$s, since otherwise $uwy \not\in L$. 
$vx$ cannot contain only $a$s, since otherwise $uv^2wx^2y \not\in L$.
none of $v$ and $x$ contains both $a$s and $b$s, since otherwise $uv^2wx^2y \not\in L$.


We conclude that $v$ contains only $a$s, $x$ contains only $b$s, and $x$ contains at least one $b$. Then, by the pumping lemma, $a^{c^2-|v|} b^{c-|x|} = uwy \in L$. This is a contradiction since:
$$
(c  - |x|)^2 \le (c - 1)^2 = c^2 - 2c + 1 \le i^2 - c &lt; i^2 - |v|.
$$