 Solution

Bloch theorem follows because lattice translations commute with $H$ and with each other: simultaneous eigenstates obey $T_R\psi =e^{ik\cdot R}\psi$, so ${\psi }_k(r)=e^{ik\cdot r}{u}_k(r)$ with $u_k$ lattice-periodic. Conversely this form has the required translation character.