  The Hammersley-Welsh bound (1962) states that the number $c_n$ of length $n$
self-avoiding walks on $\mathbb{Z}^d$ satisfies \[ c_n \leq \exp \left[
O(n^{1/2}) \right] \mu_c^n, \] where $\mu_c=\mu_c(d)$ is the connective
constant of $\mathbb{Z}^d$. While stronger estimates have subsequently been
proven for $d\geq 3$, for $d=2$ this has remained the best rigorous,
unconditional bound available. In this note, we give a new, simplified proof of
this bound, which does not rely on the combinatorial analysis of unfolding. We
also prove a small, non-quantitative improvement to the bound, namely \[ c_n
\leq \exp\left[ o(n^{1/2})\right] \mu_c^n. \] The improved bound is obtained as
a corollary to the sub-ballisticity theorem of Duminil-Copin and Hammond
(2013). We also show that any quantitative form of that theorem would yield a
corresponding quantitative improvement to the Hammersley-Welsh bound.
