  Several authors have recently proved results which express cusp forms as
$p$-adic limits of weakly holomorphic modular forms under repeated application
of Atkin's $U$-operator. The proofs involve techniques from the theory of weak
harmonic Maass forms, and in particular a result of Guerzhoy, Kent, and Ono on
the $p$-adic coupling of mock modular forms and their shadows. Here we obtain
strengthened versions of these results using techniques from the theory of
holomorphic modular forms.
