= Solution
Insert the exact solution into the <linear multistep method> and expand about $t_n$. Its defect is
$$
\sum_{l=0}^s\rho_l y(t_n+lh)-h\sum_{l=0}^s\sigma_l y'(t_n+lh).
$$
The coefficient of $h^k y^{(k)}(t_n)$ is
$$
\frac1{k!}\sum_l\rho_l l^k-\frac1{(k-1)!}\sum_l\sigma_l l^{k-1}
$$
for $k\geq1$, while the constant coefficient is $\sum_l\rho_l$. The method has order at least $p$ exactly when these coefficients vanish for $0\leq k\leq p$.
On the other hand,
$$
\rho(e^z)-z\sigma(e^z)
=\sum_l\rho_l e^{lz}-z\sum_l\sigma_l e^{lz}.
$$
The coefficient of $z^k$ in this expression is precisely the preceding order-condition coefficient. Therefore all coefficients through degree $p$ vanish exactly when
$$
\boxed{\rho(e^z)-z\sigma(e^z)=O(z^{p+1})},
$$
which proves the equivalence.
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