 Solution

Insert the exact solution into the linear multistep method and expand about $t_n$. Its defect isThe coefficient of $h^k{y}^{(k)}(t_n)$ isfor $k\ge 1$, while the constant coefficient is ${\sum }_l{\rho }_l$. The method has order at least $p$ exactly when these coefficients vanish for $0\le k\le p$.

On the other hand,The coefficient of $z^k$ in this expression is precisely the preceding order-condition coefficient. Therefore all coefficients through degree $p$ vanish exactly whenwhich proves the equivalence.