  For absolutely convergent series we state explicitly a one-sided summation
estimate that can be viewed as the discrete analogue of the change of variable
formula on the half line. This estimate is implicit in Pascal Lef\`evre's
recent elegant proof of the classical discrete Hardy inequality. Here we remove
a superfluous irrationality condition therein and point out the change of
variable character of his approach. This leads to a simpler, shorter and
\textit{bona fide} Ingham type proof of the discrete Hardy inequality, and also
provides the optimal constant.
