  We study the properties of the Resonance-Spectrum Expansion near threshold in
I=0 S-wave pion-pion scattering. The real part of the amplitude, when
extrapolated from above threshold to below threshold, is found to vanish at a
positive non-zero value of the total invariant mass of the system, in agreement
with predictions from perturbative chiral models. In our exact analytic
expression, the total amplitude vanishes identically at zero invariant mass.
