$\gamma_p(t)$ denote the outward normal geodesic emanating from $Y$ at $p$. We now show that ${\overline{E}}$ is surjective by showing that $\gamma_p$ is asymptotic to $\sigma$.

${\overline{E}}$ is surjective. \[prop:Ebaronto\]

Given the untrapped geodesic ray $\sigma$ above, the previous lemma establishes the existence of a candidate normal geodesic $\gamma_p$ to represent the equivalence class $[\sigma]$ in ${M(\infty)}$. We prove that $d( \gamma_p(t), \sigma(t) )$ remains bounded as $t {\rightarrow}\infty$. The triangle inequality implies that it is sufficient to show separately that $$d\big( (p, \sigma_r(t) ), (\sigma_Y(t), \sigma_r(t) ) \big) \; \text{and} \; d \big( (p, t), (p, \sigma_r(t)) \big)$$ remain bounded as functions of time.

*Step 1.* *$d( (p, t), (p, \sigma_r(t)) )$ is bounded as $t {\rightarrow}+\infty$:* In this situation, $d( (p, t), (p, \sigma_r(t)) ) = |t - \sigma_r(t)|$, so we must show the quantity $|t - \sigma_r(t)|$ is bounded as $t {\rightarrow}\infty$. By the fundamental theorem of calculus this is equivalent to the statement $1 - \dot\sigma_r(t) \in L^1(t_0, \infty)$ for $t_0$ sufficiently large: $$|t - \sigma_r(t)| \leq \int^{t}_{t_0} 1 - \dot\sigma_r(s) ds + |t_0 - \sigma_r(t_0)|,$$ Since $\dot\sigma_r^2 + |\dot\sigma_Y|^2 = 1$, the integrability of $1 - \dot\sigma_r(t)$ is related to the integrability of $|\dot\sigma_Y|^2$, for 