  We use epitaxial strain to shift the energy of the two-dimensional Ag(111)
surface states of Ag islands on Nb(110) substrates, allowing to spatially tune
the bottom of the surface-state band $E_{\rm SS}$ through the Fermi level
$E_{\rm F}$. Bulk and surface-state contributions to the Ag(111) local density
of states (LDOS) can be separated with scanning tunneling spectroscopy. For
thick islands ($\approx$\, 20 nm), the Ag surface states are decoupled from the
Ag bulk states via orthogonality, and the superconductive gap induced by
proximity to Nb is due to bulk states only. However, for thin islands (3-4 nm),
surface-state electrons develop superconducting correlations as identified by a
complete energy gap in the LDOS when $E_{\rm F}$ is smaller than but close to
$E_{\rm F}$. The induced superconductivity in this case is of two-band nature
and appears to occur when the surface-state wave function reaches down to the
Ag/Nb interface.
