Need a hint for a proof using the pumping lemma that a language is not regular

Question

Currently I am stuck at a proof of :

$A_2=\{w001;|w|_0<|w|_1 \wedge w\in \sum^*\}$

unsing only the pumping lemma. Can you give me a hint for a good start? thanks in advance.

Answer 1

Hint:

Assuming that $|w|_0<|w|_1$ means that the number of one's $>$ number of zero's

$$A_2=\{w001;|w|_0<|w|_1 \wedge w\in \small\sum^{\quad *} \}$$

Let us choose the word $z=1^p$ so $1^p001\in A_2$

Spoiler:

$\quad|z|>p$ So $z=uvw,\quad |uv|\leq p, \quad |v| \geq 1, \quad uv^iw\in \mathcal L \quad i\geq0$ For $i=0$ the word not in $\mathcal L$ because the number of one's$=$number of zero's