Let $X$ be a metric space and $r_1,r_2$ be two geodesic rays.
Definition. We say $r_1$ is asymptotic to $r_2$ if there exists $\alpha >0$ such that $|r_1(t)-r_2(t)|\leqslant \alpha$ for all $t\geqslant 0$ .
Definition’. We say $r_1$ is w-asymptotic to $r_2$ if $|r_1(t)-r_2(t)|=o(t)$ as $t\to \infty$.
Here are some properties.
Proposition 1. $r_1,r_2$ are asymptotic iff their image are of finite Hausdorff distance.
Suppose $X$ is moreover a Busemann space.
Proposition 2. Suppose $r_1,r_2$ start at $p\in X$, then the function $f(t)=|r_1(t)-r_2(t)|$ is increasing and tends to infinity as $t\to\infty$.
This follows directly from the convexity of $f(t)$.
Proposition 3. Suppose $r_1,r_2$ start at $p\in X$, then the followings are equivalent:
(1) $r_1=r_2$;
(2) $r_1,r_2$ are asymptotic;
(3) $r_1,r_2$ are w-asymptotic;
(4) $d_H(\operatorname{Im}(r_1),\operatorname{Im}(r_2))<\infty$.
Proof. (1) $\Leftrightarrow$ (4) is from Prop 1, and (1) $\Rightarrow$ (2) $\Rightarrow$ (3) is obvious, so only need to show (3) $\Rightarrow$ (1).
From the property of a Busemann space,
$$ f(2^kt)\leqslant \frac{1}{2}f(2^{k+1}t). $$
Thus $0=\lim_{t\to\infty}\frac{f(t)}{t}=\lim_{k\to\infty}\frac{f(2^kt_0)}{2^kt_0}\geqslant \frac{f(t_0)}{t_0}$ for any $t_0>0$, which means $r_1=r_2$.
Remark. One can also prove via: if a convex function $\R\to \R_+$ is $o(t)$ as $t\to\infty$, then it must be a constant.
Proposition 4. Let $X$ be a proper Busemann space, $r$ a geodesic ray, $q\in X$. Then there exists a unique geodesic ray $r'$ starting from $q$, which is asymptotic to $r$.
The uniqueness follows from Prop 3, and for existence, we use the Arzela-Ascoli theorem to construct $r'$.
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