  Certain smoothing inequalities were proposed in the recent paper posted on
arXiv at arxiv:1301.2828 in order to lessen the very large gap between the best
correctly established upper and lower bounds on the constant factor in the
nonuniform Berry-Esseen bound. As an illustration of the possible uses of those
inequalities, a quick proof of Nagaev's classical nonuniform bound was given
there. Here we describe another, apparently more effective class of smoothing
inequalities, and give a yet quicker and simpler proof of Nagaev's result.
