  Let $\eta_i$, $i\ge 1$, be a sequence of independent and identically
distributed random variables with finite third moment, and let $\Delta_n$ be
the total variation distance between the distribution of
$S_n:=\sum_{i=1}^n\eta_i$ and the normal distribution with the same mean and
variance. In this note, we show the dichotomy that either $\Delta_n=1$ for all
$n$ or $\Delta_n=O\left(n^{-1/2}\right)$.
