  The chain complexes underlying Floer homology theories typically carry a
real-valued filtration, allowing one to associate to each Floer homology class
a spectral number defined as the infimum of the filtration levels of chains
representing that class. These spectral numbers have been studied extensively
in the case of Hamiltonian Floer homology by Oh, Schwarz, and others. We prove
that the spectral number associated to any nonzero Floer homology class is
always finite, and that the infimum in the definition of the spectral number is
always attained. In the Hamiltonian case, this implies that what is known as
the ``nondegenerate spectrality'' axiom holds on all closed symplectic
manifolds. Our proofs are entirely algebraic and apply to any Floer-type theory
(including Novikov homology) satisfying certain standard formal properties. The
key ingredient is a theorem about the existence of best approximations of
arbitrary elements of finitely generated free modules over Novikov rings by
elements of prescribed submodules with respect to a certain family of
non-Archimedean metrics.
