  In this paper, we prove the linear stability to gravitational and
electromagnetic perturbations of the Reissner-Nordstr\"om family of charged
black holes with small charge. Solutions to the linearized Einstein-Maxwell
equations around a Reissner-Nordstr\"om solution arising from regular initial
data remain globally bounded on the black hole exterior and in fact decay to a
linearized Kerr-Newman metric. We express the perturbations in geodesic
outgoing null foliations, also known as Bondi gauge. To obtain decay of the
solution, one must add a residual pure gauge solution which is proved to be
itself controlled from initial data. Our results rely on decay statements for
the Teukolsky system of spin $\pm2$ and spin $\pm1$ satisfied by
gauge-invariant null-decomposed curvature components, obtained in earlier
works. These decays are then exploited to obtain polynomial decay for all the
remaining components of curvature, electromagnetic tensor and Ricci
coefficients. In particular, the obtained decay is optimal in the sense that it
is the one which is expected to hold in the non-linear stability problem.
