Describe the error in the following $$proof$$ that $01$ is not a regular language

$($an error must exist because $01$ is regular$).$ The proof is by contradiction.

$$ Assume $01$ is regular.

$$ Let p be the pumping length for $01$ given by the pumping lemma.

$$ Choose s to be the string $0$p$1$p$.$

$$ We know that s is a member of $01,$ but a previous proof$$ shows that s cannot

be pumped.

$$ Thus there is a contradiction. So $01$ is not regular.

$$This is demonstrated both in example $1\mathrm{.}73$ in your textbook and in example $1$ of the

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