Proof of Bose condensation for weakly interacting lattice bosons

A weakly interacting Bose gas on a simple cubic lattice is considered. We prove the existence of the standard or zero-mode Bose condensation at sufficiently low temperature. This result is valid for sufficiently small interaction potential and small values of chemical potential. Our method exploits infrared bound for the suitable two-point Bogolyubov's inner product. We do not use the reflection positivity or some expansion methods.