Proof: A single-step method for approximating the solution of the IVP

is given by

(5.2)

A change in one of the computed values will lead us to solve

(5.3)

instead of (5.2).

Subtracting (5.3) from (5.2), we get

Since satisfies a Lipschitz condition in , we get

which is a bounded multiple of the introduced error and is independent of . Hence the given method (5.1) is stable provided satisfies a Lipschitz c ondition in .