  Let $(x_n)$ be a positive real sequence decreasing to $0$ such that the
series $\sum_n x_n$ is divergent and $\liminf_{n} x_{n+1}/x_n>1/2$. We show
that there exists a constant $\theta \in (0,1)$ such that, for each $\ell>0$,
there is a subsequence $(x_{n_k})$ for which $\sum_k x_{n_k}=\ell$ and
$x_{n_k}=O(\theta^k)$.
