= Lipschitz one-step error bound
{c}
{title2=$|e_n|\leq(1+hK)^n|e_0|+\frac{(1+hK)^n-1}{K}\max_{j<n}|T_j|$}
Suppose the increment function of a <one-step method> is <Lipschitz continuous> in its solution argument with constant $K>0$, uniformly in time and step size. Let $e_n=y(x_n)-y_n$ be the <global error>, and let $hT_n$ be the exact-start <local truncation error>. The <triangle inequality> gives $|e_{n+1}|\leq(1+hK)|e_n|+h|T_n|$. Iterating and summing a <geometric series> proves the displayed estimate. Since $(1+hK)^n\leq e^{Knh}$, uniformly bounded $K$ and <consistency error> $O(h^p)$ give <global error> $O(h^p)$ on a fixed time interval when $e_0=O(h^p)$. Here $O$ denotes <Big O notation>. This is a constant-step form of the <discrete Gronwall inequality>.
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