Some comments on a theorem of Hardy and Littlewood

In this note, we reconstruct a proof of a classical result due to Hardy and Littlewood. While this result has played an important role in the modern theories of Markov decision processes and stochastic games, it is not that easy to find its proof in the literature in the format in which it has been applied. Furthermore, we supply either examples or complete citations for the other related cases which are not covered by the Hardy-Littlewood theorem.