Translation-character proof of Bloch theorem

The lattice translations commute with one another and with a lattice-periodic Hamiltonian. They can therefore be diagonalized simultaneously within each energy eigenspace. Every unitary character of a Bravais lattice has the form $r\mapsto e^{-ik\cdot r}$, with $k$ defined modulo the reciprocal lattice. Hence $T_r{\psi }_k=e^{-ik\cdot r}{\psi }_k$, so $u_k(x)=e^{-ik\cdot x}{\psi }_k(x)$ is lattice-periodic.