  The spectral propinquity is a distance, up to unitary equivalence, on the
class of metric spectral triples. We prove in this paper that if a sequence of
metric spectral triples converges for the propinquity, then the spectra of the
Dirac operators for these triples do converge to the spectrum of the Dirac
operator at the limit. We obtain this result by first proving that, in an
appropriate sense induced by some natural metric, the bounded, continuous
functional calculi defined by the Dirac operators also converge. As an
application of our work, we see, in particular, that action functionals of a
wide class of metric spectral triples are continuous for the spectral
propinquity, which clearly connects convergence for the spectral propinquity
with the applications of noncommutative geometry to mathematical physics. This
fact is a consequence of results on the continuity of the multiplicity of
eigenvalues of Dirac operators. In particular, we formalize convergence of
adjoinable operators of different C*-correspondences, endowed with appropriate
quantum metric data.
