  Let $S_{n,k}$ denote the random geometric graph obtained by placing points in
a square box of area $n$ according to a Poisson process of intensity 1 and
joining each point to its $k$ nearest neighbours. Balister, Bollob\'as, Sarkar
and Walters conjectured that for every $0< \epsilon <1$ and all $n$
sufficiently large there exists $C=C(\epsilon)$ such that whenever the
probability $S_{n,k}$ is connected is at least $\epsilon $ then the probability
$S_{n,k+C}$ is connected is at least $1-\epsilon $. In this paper we prove this
conjecture.
  As a corollary we prove that there is a constant $C'$ such that whenever
$k=k(n)$ is a sequence of integers such that the probability $S_{n,k(n)}$ is
connected tends to one as $n$ tends to infinity, then for any $s(n)$ with
$s(n)=o(\log n)$, the probability that $S_{n,k(n)+C's\log \log n}$ is
$s$-connected tends to one This proves another conjecture of Balister,
Bollob\'as, Sarkar and Walters.
