  We ask whether the Eigenstate Thermalization Hypothesis (ETH) is valid in a
strong sense: in the limit of an infinite system, {\it every} eigenstate is
thermal. We examine expectation values of few-body operators in highly-excited
many-body eigenstates and search for `outliers', the eigenstates that deviate
the most from ETH. We use exact diagonalization of two one-dimensional
nonintegrable models: a quantum Ising chain with transverse and longitudinal
fields, and hard-core bosons at half-filling with nearest- and
next-nearest-neighbor hopping and interaction. We show that even the most
extreme outliers appear to obey ETH as the system size increases, and thus
provide numerical evidences that support ETH in this strong sense. Finally,
periodically driving the Ising Hamiltonian, we show that the eigenstates of the
corresponding Floquet operator obey ETH even more closely. We attribute this
better thermalization to removing the constraint of conservation of the total
energy.
