= Solution
<Convergence of a numerical method> on a fixed interval $[0,T]$ means that, as the step size tends to zero and the starting approximations tend to the exact initial values,
$$
\boxed{\max_{0\leq t_n\leq T}|y_n-y(t_n)|\longrightarrow0.}
$$
For a vector-valued <ordinary differential equation>, replace the absolute value by a <norm>. The interval remains fixed as $h\to0$; a small error at only one step is insufficient.
The <order of a numerical method> is measured by its defect on a smooth exact solution. With the unscaled residual convention for a <linear multistep method>, order at least $p$ means local residual $O(h^{p+1})$; equivalently, the residual divided by $h$ is $O(h^p)$. Order exactly $p$ means the next coefficient does not vanish in general. For a <zero-stable> method, under the usual smoothness and <Lipschitz continuity> assumptions and with starting errors $O(h^p)$, this yields <global error> $O(h^p)$ on the fixed interval. Local order by itself does not imply convergence without stability.
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