  Recently, a new notion of quantum R\'enyi divergences has been introduced by
M\"uller-Lennert, Dupuis, Szehr, Fehr and Tomamichel, J.Math.Phys. 54:122203,
(2013), and Wilde, Winter, Yang, Commun.Math.Phys. 331:593--622, (2014), that
has found a number of applications in strong converse theorems. Here we show
that these new R\'enyi divergences are also useful tools to obtain coding
theorems in the direct domain of various problems. We demonstrate this by
giving new and considerably simplified proofs for the achievability parts of
Stein's lemma with composite null hypothesis, universal state compression, and
the classical capacity of compound classical-quantum channels, based on
single-shot error bounds already available in the literature, and simple
properties of the quantum R\'enyi divergences. The novelty of our proofs is
that the composite/compound coding theorems can be almost directly obtained
from the single-shot error bounds, with essentially the same effort as for the
case of simple null-hypothesis/single source/single channel.
