= Solution
Insert a smooth exact function into the <multistep method> and expand its residual about $t_n$. Define
$$
c_0=\sum_{i=0}^s\rho_i,\qquad c_k=\sum_{i=0}^s\rho_i i^k-k\sum_{i=0}^s\sigma_i i^{k-1}\quad(k\ge1).
$$
The residual is $\sum_{k\ge0}h^kc_k y^{(k)}(t_n)/k!$. Therefore order at least $p$ is equivalent to $c_0=\cdots=c_p=0$. Applying the residual to the monomials $(t-t_n)^k$, $k\le p$, gives exactly those conditions. Linearity then makes them equivalent to exactness on every polynomial of degree at most $p$, as asserted. For exact rather than at-least order $p$, the next coefficient must not vanish.
If an $s$-step formula were exact for degree $2s+1$, use <Hermite interpolation> to choose $Q$ with $Q(t_{n+i})=0$ for $i<s$, $Q(t_{n+s})=1$, and $Q'(t_{n+i})=0$ for every $i$. Its degree is at most $2s+1$. The purported identity would then read $\rho_s=0$, contradicting $\rho_s=1$. \b[No $s$-step method has order $2s+1$ or higher.]
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