We investigate a possibility of superfluidity in a trapped gas of Fermi atoms with repulsive interaction U in the presence of an optical lattice. Applying the exact diagonalization method to a one-dimensional fermion-Hubbard model including the trap potential, we find that, when U exceeds a critical value, the binding energy of two Fermi atoms becomes negative indicating that an attractive interaction effectively works. In this case, a "Mott core" appears in the trap center, where each site is occupied by one atom, and a large Cooper-pair amplitude develops between atoms in the left and right hand sides of this core. In order to clarify the origin of the attractive force, we compute a pseudo spin correlation between neighboring sites given by 〈SiSi+1〉. As a result, we find that a creation of a singlet pair between both sides of the Mott core is strongly coupled with a pseudo-spin singlet re-formation leading to an energy gain process.