  We present a lattice model for superconducting fermions whose
nearest-neighbour two-body interactions are a Bardeen-Cooper-Schrieffer-type
pairing on the hypercubic lattice $\mathbb{Z}^d$ with the dimension $d\ge 3$.
Although these effective interactions between two electrons are believed to be
caused by electron-phonon interactions, we assume that the interactions between
two electrons are of short range without phonons. For the model, we prove the
existence of the long-range order of the superconductivity at low temperatures,
and also prove the existence of a gapless excitation above the infinite-volume
ground state. Namely, there appears a Nambu-Goldstone mode which is associated
with the U(1) symmetry breaking. The corresponding Nambu-Goldstone boson is
exactly a Cooper pair, which consists of spin up and down electrons.
