  In this note we prove a sharp lower bound on the necessary number of nestings
of nested absolute-value functions of generalized hinging hyperplanes (GHH) to
represent arbitrary CPWL functions. Previous upper bound states that $n+1$
nestings is sufficient for GHH to achieve universal representation power, but
the corresponding lower bound was unknown. We prove that $n$ nestings is
necessary for universal representation power, which provides an almost tight
lower bound. We also show that one-hidden-layer neural networks don't have
universal approximation power over the whole domain. The analysis is based on a
key lemma showing that any finite sum of periodic functions is either
non-integrable or the zero function, which might be of independent interest.
