  A Hopf algebra is co-Frobenius when it has a nonzero integral. It is proved
that the composition length of the indecomposable injective comodules over a
co-Frobenius Hopf algebra is bounded. As a consequence, the coradical
filtration of a co-Frobenius Hopf algebra is finite; this confirms a conjecture
by Sorin D\u{a}sc\u{a}lescu and the first author. The proof is of categorical
nature and the same result is obtained for Frobenius tensor categories of
subexponential growth. A family of co-Frobenius Hopf algebras that are not of
finite type over their Hopf socles is constructed, answering so in the negative
another question by the same authors.
