  We prove the equivalence between the integral central limit theorem and the
local central limit theorem for two-body potentials with long-range
interactions on the lattice $\mathbb{Z}^d$ for $d\ge 1$. The spin space can be
an arbitrary, possibly unbounded subset of the real axis with a suitable
a-priori measure. For general unbounded spins, our method works at high-enough
temperature, but for bounded spins our results hold for every temperature. Our
proof relies on the control of the integrated characteristic function, which is
achieved by dividing the integration into three different regions, following a
standard approach proposed forty years ago by Campanino, Del Grosso and
Tirozzi. The bounds required in the different regions are obtained through
cluster-expansion techniques. For bounded spins, the arbitrariness of the
temperature is achieved through a decimation ("dilution") technique, also
introduced in the later reference.
