  We consider solutions of the scalar wave equation $\Box_g\phi=0$, without
symmetry, on fixed subextremal Reissner-Nordstr\"om backgrounds $({\mathcal M},
g)$ with nonvanishing charge. Previously, it has been shown that for $\phi$
arising from sufficiently regular data on a two ended Cauchy hypersurface, the
solution and its derivatives decay suitably fast on the event horizon
${\mathcal H}^+$. Using this, we show here that $\phi$ is in fact uniformly
bounded, $|\phi| \leq C$, in the black hole interior up to and including the
bifurcate Cauchy horizon ${\mathcal C}{\mathcal H}^+$, to which $\phi$ in fact
extends continuously. The proof depends on novel weighted energy estimates in
the black hole interior which, in combination with commutation by angular
momentum operators and application of Sobolev embedding, yield uniform
pointwise estimates. In a forthcoming companion paper we will extend the result
to subextremal Kerr backgrounds with nonvanishing rotation.
