I am reading the book 'Metric spaces of non-positive curvature' by Bridson and Haefliger. In page 427 the following is said:
Two geodesic rays $c, c^\prime \colon [0, \infty)\rightarrow X$ are said to be asymptotic if $\sup_{t} d(c(t),c^\prime(t))$ is finite; this condition is equivalent to saying that the Hausdorff distance between the images of $c$ and $c^\prime$ is finite.
The Hausdorff distance of general subsets $A,B$ is defined as follows: $d_{H}(A,B)=\inf\{ \epsilon \mid A \subseteq N_{\epsilon}(B), B\subseteq N_{\epsilon}(A)\}$
It is clear that if $\sup_{t} d(c(t),c^\prime(t))$ is finite, the Hausdorff distance is finite.
Now assume that the Hausdorff distance is finite (say $k$) and let $t\in [0, \infty)$. Then, there exists $t^\prime \in [0, \infty)$ such that $d(c(t), c^\prime(t^\prime)) \leq k$. Hence, $d(c(t), c^\prime(t)) \leq d(c(t), c^\prime(t^\prime))+ d(c^\prime(t^\prime), c^\prime(t))\leq k+d(c^\prime(t^\prime),c^\prime(t))=k+|t-t^\prime|$.
I should find an upper bound independent of $t$ for that number, but I do not know how to proceed. Can anyone help me? Thanks in advance.
Suppose the Hausdorff distance is bounded above by $K<\infty$.
Let $C_0=|c(0)-c'(0)|$.
Fix $t\in \mathbb{R}$, we want to bound $|c(t)-c'(t)|$. Let $t'$ be a point for which $|c(t)-c'(t')|\leq K$.
We know that the distance from $c'(0)$ to $c'(t')$ is $t'$ because $c'$ is a geodesic.
Alternatively, we can jump from $c'(t')$ to $c(t)$, and from $c(t)$ to $c(0)$, and then finally from $c(0)$ to $c'(0)$. The total length of this path is bounded by $K+t+C_0$.
We conclude that $t'\leq K+t+C_0$ (Formally, I am using the triangle inequality twice here).
A simlar argument shows that $t\leq K+t'+C_0$.
So $|t-t'| \leq K+C_0$.
Combine this what you've already written above.