  The average properties of the well-known Subset Sum Problem can be studied by
the means of its randomised version, where we are given a target value $z$,
random variables $X_1, \ldots, X_n$, and an error parameter $\varepsilon > 0$,
and we seek a subset of the $X_i$s whose sum approximates $z$ up to error
$\varepsilon$. In this setup, it has been shown that, under mild assumptions on
the distribution of the random variables, a sample of size
$\mathcal{O}(\log(1/\varepsilon))$ suffices to obtain, with high probability,
approximations for all values in $[-1/2, 1/2]$. Recently, this result has been
rediscovered outside the algorithms community, enabling meaningful progress in
other fields. In this work we present an alternative proof for this theorem,
with a more direct approach and resourcing to more elementary tools.
