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For a fixed-coefficient linear multistep method applied to a well-posed initial value problem with a continuous right-hand side uniformly Lipschitz continuous in the solution argument on the relevant region, consistency of a numerical method and zero-stability are equivalent to numerical convergence for arbitrary starting values whose errors tend to zero. The time interval is fixed while the step size tends to zero; an implicit numerical method must use its well-defined nearby solution branch for sufficiently small steps.

The quantitative statement is that a zero-stable method with consistency error $O(h^p)$ and starting errors $O(h^p)$ has global error $O(h^p)$, provided the exact solution is sufficiently smooth for that residual bound. A bounded response to step defects follows from the root condition for a multistep method, and the discrete Gronwall inequality controls the nonlinear error terms. Mere continuity of the right-hand side does not by itself supply these hypotheses or uniqueness of the exact solution.

An explicit numerical method normally has a simple, inexpensive step and needs no simultaneous solve, but stability can force very small steps, especially for stiff differential equations. An implicit numerical method requires a linear or nonlinear solve, storage and suitable solver tolerances; its larger stability region can permit larger steps when stability is the main restriction. Accuracy, phase fidelity and preservation of physical structure still constrain the step. For this oscillator, backward Euler method is stable but dissipative, while forward Euler method is cheap but unstable without damping. Neither statement makes all implicit methods superior to all explicit ones; method order, problem structure and total cost determine the useful choice.