= Cayley coefficient proof of the first Dahlquist barrier
{c}
{title2=$p\leq s+1\ (s\text{ odd}),\quad p\leq s+2\ (s\text{ even})$}
For a real consistent <zero-stable> <linear multistep method>, transform its first <characteristic polynomial> by $z=(1+x)/(1-x)$ to $(1-x)^s\rho(z)=xP(x)$. The <root condition for a multistep method> puts all finite roots of $P$ in the closed left half-plane; real linear factors and conjugate quadratic factors consequently give nonnegative coefficients, after choosing $P(0)>0$. The order criterion requires a <polynomial> of degree at most $s$ to agree through degree $p-1$ with $P(x)x/(2\operatorname{arctanh}x)$. The latter quotient has strictly negative nonconstant even coefficients. Its first even coefficient beyond degree $s$ includes a strictly negative multiple of $P(0)$ and cannot vanish. This gives the stated bound.
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