Show a language is not regular by using the pumping lemma

Question

Show the following language is not regular by using the pumping lemma

$A_1=\{0^n1^n2^n|n \geq 0\}$

Proof by Contradiction:

Assume $A_1$ is Regular.

Let M be the pumping length and let S be a string in $A_1$ such that $|S|\geq M$, so let $$S=0^M1^M2^M$$

Because of the pumping lemma, we can divide S into three pieces xyz, so let $S=xyz$ where $|y|>0$ and $|xy|\leq M$.

Let $xy=0^M$ and $z=1^M2^M$, so we can say $x=0^{M-1}$ and $y=0$.

Since the pumping lemma says that any $xy^iz\in A$ where $i\geq 0$, then $xy^2z\in A$. Therefore,

$$0^{M-1}0^21^M2^M=0^{M+1}1^M2^M\in A_1$$

Which is a contradiction to the original language. Therefore, $A_1$ is not regular.

Ho does that look?. This is my first proof with the pumping lemma.

Answer 1

Looks good otherwise, except you need to show that there is no eligible $xyz$ division. You've only shown that $x = 0^{M-1}$ and $y = 0$ is not eligible. Modifying the proof isn't hard though, just let $x = 0^{M-n-k}, y = 0^n, z = 0^k 1^M 2^M$ where $M \geq n > 0$.