A note on cusp forms as p-adic limits

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.