I need to show that $${0^m1^n} text{ with } m neq n$$ is not regular.

I tried to do this using the pumping lemma. Take $s=0^{d_1} 1^{d_2}$ with $d_1 neq d_2$. Given a decomposition $s=xyz$, there are $3$ possibilities.

$1.$ $y$ contains only $0$‘s.

If $d_1 < d_2$ and $|y|=1$, $d_1$ will eventually be equal to $d_2$, which is not accepted by $L$.

$2.$ $y$ contains only $1$‘s.

If $d_2 < d_1$ and $|y|=1$, $d_1$ will eventually be equal to $d_2$, which is not accepted by $L$.

$3.$ $y$ has the form $(0)^+(1)^+$

We illustrate this with an example. If we take $xyz$, then $xyyz$ will have a $1$ flowed inside the row’s of $0$‘s and that is not accepted by $L$ as first we have all $0$‘s and then $1$‘s.

Consequently, $L$ cannot be regular.