Describe the error in the following "proof" that $0*1*$ is not a regular language. $($An error must exist because $0*1*$ is regular.$)$

The proof is by contradiction. Assume that $0*1*$ is regular. Let p be the pumping length for $0*1*$ given by the pumping lemma. Choose s to be the string $0$p$1$p$.$ You know that s is a member of $0*1*,$ but Example $1\mathrm{.}73$ in the book shows that s cannot be pumped. Thus you have a contradiction. So $0*1*$ is not regular.