We derive rigorous bounds on the average momentum occupation numbers $langle n_{mathbf{k}sigma}rangle$ in the Hubbard and Kondo models in the ground state and at non-zero temperature ($T>0$) in the grand canonical ensemble. For the Hubbard model with $T>0$ our bound proves that, when interaction strength $ll k_B Tll$ Fermi energy, $langle n_{mathbf{k}sigma}rangle$ is guaranteed to be close to its value in a low temperature free fermion system. For the Kondo model with any $T>0$ our bound proves that $langle n_{mathbf{k}sigma}rangle$ tends to its non-interacting value in the infinite volume limit. In the ground state case our bounds instead show that $langle n_{mathbf{k}sigma}rangle$ approaches its non-interacting value as $mathbf{k}$ moves away from a certain surface in momentum space. For the Hubbard model at half-filling on a bipartite lattice, this surface coincides with the non-interacting Fermi surface. In the Supplemental Material we extend our results to some generaliz
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