Simple proof that an adiabatic invariant is conserved to exponential accuracy over a complete interval of development

The Bogolyubov–Zubarev fast-phase method [1] is used to give a simple proof of the following well-known proposition [2]: when a perturbation is switched on and off smoothly, the total increment of the adiabatic invariant (the action) for one-dimensional periodic motion in a slowly varying potential is a quantity less than any power of $\alpha$, where $\alpha$ is a parameter that characterizes the rate of change of the potential.