Show that, in the Tight Binding approximation (in an arbitrary crystal lattice with an arbitrary number of neighbors retained), the band energy E(n)(k) satisfies
E(n)(k) = E(n)(k + G)
i.e. the energy has the periodicity of the reciprocal lattice, so it is sufficient, for each band n, to know E(n)(k) for k in the first Brillouin Zone.