Geodesic ray converges to infinity

Question

I am reading this paper on boundaries of hyperbolic groups. In this paper, a geodesic metric space $(X,d)$ is considered. A sequence of points $(x_n)_{n \geq 1}$ converges to infinity if $$\lim \inf_{i,j \to \infty} (x_i, x_j)_x = \infty$$ where $(x_i, x_j)_x$ is the Gromov product with respect to the point $x$, that is $$(x_i,x_j)_x = \frac{1}{2}(d(x_i,x) + d(x_j,x) - d(x_i,x_j))$$

The paper then states that for a geodesic ray $\gamma: [0, + \infty) \to X$ the sequence $\gamma(n)$ converges to infinity, which I tried to prove, but which I couldn't prove.

Any help would be appreciated!

EDIT: Using @Seirios' hint, I came up with the following reasoning: We first compute $(\gamma(i),\gamma(j))_{\gamma(0)}$. Assuming $i \leq j$, we find that $$(\gamma(i),\gamma(j))_{\gamma(0)} = \frac{1}{2}(d(\gamma(i), \gamma(0)) + d(\gamma(j), \gamma(0)) - d(\gamma(i), \gamma(j)))$$ and since $\gamma$ is a geodesic ray, we have that $d(\gamma(t), \gamma(t')) = |t - t'|$, hence the above equation simplifies to $$(\gamma(i),\gamma(j))_{\gamma(0)} = \frac{1}{2}(i + j - j + i) = i.$$

By Seirios' second hint, we have that for any $x \in X$ $$(\gamma(i),\gamma(j))_{\gamma(0)} - d(\gamma(0),x) \leq (\gamma(i),\gamma(j))_x$$ where we can assume $d(\gamma(0),x)$ to be finite (since the limit notion does not depend on $x$. Taking limes inferior on both sides shows that the left hand side goes to $\infty$ and hence the right hand side must too.

Answer 1

Hint 1: First compute $\left( \gamma(i),\gamma(j) \right)_{\gamma(0)}$, and next compare $\left( \gamma(i),\gamma(j) \right)_{\gamma(0)}$ and $\left( \gamma(i),\gamma(j) \right)_{x}$ for an arbitrary $x \in X$.

Hint 2: Notice that, for every $x,y,u,v \in X$, one has $(x,y)_u \leq (x,y)_v + d(u,v)$.

The intuition behind the Gromov product is that $(y,z)_x$ quantifies the amount of time two geodesics $[x,y]$ and $[x,z]$ fellow travel. To convince yourself:

• compute the Gromov product of three vertices in a tree;
• in a $\delta$-hyperbolic space $X$, given three points $x,y,z \in X$ and two geodesics $[x,y]$ and $[x,z]$, compute the distance between $y_t$ and $z_t$ for every $t \leq (y,z)_x$, where $y_t$ and $z_t$ denote the points of $[x,y]$ and $[x,z]$ respectively at distance $t$ from $x$.