In an isolated quantum many-body system undergoing unitary evolution, we
study the thermalization of a subsystem, treating the rest of the system as a
bath. In this setting, the eigenstate thermalization hypothesis (ETH) was
proposed to explain thermalization. Consider a nearly integrable
Sachdev-Ye-Kitaev model obtained by adding random all-to-all $4$-body
interactions as a perturbation to a random free-fermion model. When the
subsystem size is larger than the square root of but is still a vanishing
fraction of the system size, we prove thermalization if the system is
initialized in a random product state, while almost all eigenstates violate the
ETH. In this sense, the ETH is not a necessary condition for thermalization.
