$Y_1, Y_2$ where $Y_i$ is the boundary of an essential subset $K_i$. We use the notation $\gamma_{p'}$ to denote the normal geodesic to $Y_1$ emanating from the point $p' \in Y_1$, and $\sigma_{q'}$ to denote the normal geodesic to $Y_2$ emanating from the point $q' \in Y_2$.

Let $K_1$ and $K_2$ be essential subsets of $M$ with $Y_i = \partial K_i$. Given any point $q \in Y_2$ there exists an open neighbourhood $V_q \subset Y_2$ of $q$ and $B > 0$ such that for every $q' \in V_q$: $$d(\sigma_{q'}(t), \gamma_{p'}(t)) \leq B, \mbox{for all } t \geq 0,$$ where $\gamma_{p'}$ is the unique normal geodesic ray emanating from $Y_1$ that is asymptotic to $\sigma_{q'}$. \[lemma:unifbdd\]

Let the exponential map ${E}\colon Y_1 \times [0, \infty) {\rightarrow}{M \backslash K}_1$ be denoted ${E}_1$.

Fix $q \in Y_2$ and $R>0$. Now $\sigma_{q}$ is eventually outside every compact set, so there exists $T \geq 0$ such that $\sigma_{q}(t) \in {E}_1(Y_1 \times [R, \infty))$ for $t\geq T$. Further the $r$-component of this curve, $(\sigma_{q})_r$, is eventually strictly monotone increasing so we may increase $T$ if necessary to ensure that $(\dot\sigma_{q})_r(t) > 0$ for $t \geq T$. By continuity of the exponential map, there is a precompact 