Lipschitz one-step error bound ($|e_n|\le (1+hK{)}^n|e_0|+\frac{(1+hK{)}^n-1}{K}{\max }_{j<n}|T_j|$)

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 $h{T}_n$ be the exact-start local truncation error. The triangle inequality gives $|e_{n+1}|\le (1+hK)|e_n|+h|T_n|$. Iterating and summing a geometric series proves the displayed estimate. Since $(1+hK{)}^n\le e^{Knh}$, 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.