  We study the proximity effect in a one-dimensional nanowire strongly coupled
to a finite superconductor with a characteristic size which is much shorter
than its coherence length. Such geometries have become increasingly relevant in
recent years in the experimental search for Majorana fermions with the
development of thin epitaxial Al shells which form a very strong contact with
either InAs or InSb nanowires. So far, however, no theoretical treatment of the
proximity effect in these systems has accounted for the finite size of the
superconducting film. We show that the finite-size effects become very
detrimental when the level spacing of the superconductor greatly exceeds its
energy gap. Without any fine-tuning of the size of the superconductor (on the
scale of the Fermi wavelength), the tunneling energy scale must be larger than
the level spacing in order to reach the hard gap regime which is seen
ubiquitously in the experiments. However, in this regime, the large tunneling
energy scale induces a large shift in the effective chemical potential of the
nanowire and pushes the topological phase transition to magnetic field
strengths which exceed the critical field of Al.
