Solution
= Solution
Write the nonzero ordinal in <Cantor normal form>
$$
\alpha=\omega^{\delta_1}n_1+\omega^{\delta_2}n_2+\cdots+\omega^{\delta_r}n_r,
$$
where $\delta_1>\cdots>\delta_r$ and $0<n_i<\omega$. Taking $\delta=\delta_1$, $n=n_1$, and
$$
\eta=\omega^{\delta_2}n_2+\cdots+\omega^{\delta_r}n_r
$$
gives $\alpha=\omega^\delta n+\eta$ with $\eta<\omega^\delta$.