  It has previously been suggested that small subsystems of closed quantum
systems thermalize under some assumptions; however, this has been rigorously
shown so far only for systems with very weak interaction between subsystems. In
this work, we give rigorous analytic results on thermalization for
translation-invariant quantum lattice systems with finite-range interaction of
arbitrary strength, in all cases where there is a unique equilibrium state at
the corresponding temperature. We clarify the physical picture by showing that
subsystems relax towards the reduction of the global Gibbs state, not the local
Gibbs state, if the initial state has close to maximal population entropy and
certain non-degeneracy conditions on the spectrum are satisfied. Moreover, we
show that almost all pure states with support on a small energy window are
locally thermal in the sense of canonical typicality. We derive our results
from a statement on equivalence of ensembles generalizing earlier results by
Lima, and give numerical and analytic finite-size bounds, relating the Ising
model to the finite de Finetti theorem. Furthermore, we prove that global
energy eigenstates are locally close to diagonal in the local energy
eigenbasis, which constitutes a part of the eigenstate thermalization
hypothesis that is valid regardless of the integrability of the model.
