  Ferromagnetic insulators can induce a strong exchange field in an adjacent
superconductor via the magnetic proximity effect. This manifests as
spin-splitting of the BCS density of states of the superconductor, an important
ingredient for numerous superconducting spintronics applications and the
realization of Majorana fermions. A crucial parameter that determines the
magnitude of the induced spin-splitting in FI/S bilayers is the thickness of
the S layer $d$: In very thin samples, the superconductivity is suppressed by
the strong magnetism. By contrast, in very thick samples, the spin splitting is
absent at distances away from the interface. In this work, we calculate the
density of states and critical field of FI/S bilayers of arbitrary thickness.
From here, we determine the range of parameters of interest for applications,
where the exchange field and superconductivity coexist. We compare our theory
with the tunneling spectroscopy measurements in several EuS/Al/AlO$_x$/Al
samples. If the Al film in contact with the EuS is thinner than a certain
critical value, we do not observe superconductivity, whereas, in thicker
samples, we find evidence of a first-order phase transition induced by an
external field. The complete transition is preceded by a regime in which normal
and superconducting regions coexist. We attribute this mixed phase to
inhomogeneities of the Al film thickness and the presence of superparamagnetic
grains at the EuS/Al interface with different switching fields. Our results
demonstrate on the one hand, the important role of the S layer thickness, which
is particularly relevant for the fabrication of high-quality samples suitable
for applications. On the other hand, the agreement between theory and
experiment demonstrates the accuracy of our theory, which, originally developed
for homogeneous situations, is generalized to highly inhomogeneous systems.
