On a problem of Simons and Johnson

This note solves a problem (related to proving a convergence which is stronger than convergence in total variation norm in the contexts like that of the celebrated Poisson convergence of binomial probabilities) stated towards the end of Simons and Johnson [1971. On the convergence of binomial to Poisson distribution. Ann. Math. Statist. 42, 1735-1736], using an approach different from theirs. The present method is simple, goes beyond the context of Poisson convergence and is expected to be more widely applicable.