  We present an accurate method of treating the one-neutron removal reaction at
intermediate incident energies induced by both nuclear and Coulomb
interactions. In the method, the nuclear and Coulomb breakup processes are
consistently treated by the method of continuum discretized coupled channels
without making the adiabatic approximation to the Coulomb interaction, so that
the removal cross section calculated never diverges. This method is applied to
recently measured one-neutron removal cross section for $^{31}$Ne+$^{12}$C
scattering at 230 MeV/nucleon and $^{31}$Ne+$^{208}$Pb scattering at 234
MeV/nucleon. The spectroscopic factor and the asymptotic normalization
coefficient of the last neutron in $^{31}$Ne are evaluated.
