Why does this Pumping Lemma example show irregularity?
$s$ consists of $p$ zeroes followed by, well, it doesn't matter what. $xy$&nbsp;is the start of&nbsp;$s$ and consists of at most the first&nbsp;$p$ characters of&nbsp;$s$. All of those $p$&nbsp;characters are zeroes so, in particular, $y$&nbsp;is all zeros.

Step&nbsp;4 of your proof is invalid because you can't choose what $x$ and&nbsp;$z$ are. The pumping lemma tells you that, if $C$&nbsp;is regular, there is some way of writing every appropriate&nbsp;$s\in C$ as $s=xyz$ such that blah, blah, blah. It doesn't tell you that writing $s=\varepsilon y\varepsilon$ will work.