 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,For a vector-valued ordinary differential equation, replace the absolute value by a norm. The interval remains fixed as $h\to 0$; 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.