  We show that, under mild assumptions on the limiting curve, a sequence of
simple chordal planar curves converges uniformly whenever certain Loewner
driving functions converge. We extend this result to random curves. The random
version applies in particular to random lattice paths that have chordal
$\mathrm {SLE}_{\kappa}$ as a scaling limit, with $\kappa <8$
(nonspace-filling). Existing $\mathrm {SLE}_{\kappa}$ convergence proofs often
begin by showing that the Loewner driving functions of these paths (viewed from
$\infty$) converge to Brownian motion. Unfortunately, this is not sufficient,
and additional arguments are required to complete the proofs. We show that
driving function convergence is sufficient if it can be established for both
parametrization directions and a generic observation point.
