  We study proximity effect of pair correlation in the inner crust of neutron
stars by means of the Skyrme-Hartree-Fock-Bogoliubov theory formulated in the
coordinate space. We describe a system composed of a nuclear cluster immersed
in neutron superfluid, which is confined in a spherical box. Using a
density-dependent effective pairing interaction which reproduces both the pair
gap of neutron matter obtained in ab initio calculations and that of finite
nuclei, we analyze how the pair condensate in neutron superfluid is affected by
the presence of the nuclear cluster. It is found that the proximity effect is
characterized by the coherence length of neutron superfluid measured from the
edge position of the nuclear cluster. The calculation predicts that the
proximity effect has a strong density dependence. In the middle layers of the
inner crust with baryon density $5 \times 10^{-4}$ fm$^{-3} < \rho_b < 2\times
10^{-2}$ fm$^{-3}$, the proximity effect is well limited in the vicinity of the
nuclear cluster, i.e. in a sufficiently smaller area than the Wigner-Seitz
cell. On the contrary, the proximity effect is predicted to extend to the whole
volume of the Wigner-Seitz cell in shallow layers of the inner crust with
$\rho_b < 2 \times 10^{-4}$ fm$^{-3}$, and in deep layers with $\rho_b > 5
\times 10^{-2}$ fm$^{-3}$.
