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I'm reading the original paper of Gromov Hyperbolic Groups. There, he gives the next example

Let $X_0,d$ be an arbitrary metric space ande let $f\colon\mathbb{R}\rightarrow \mathbb{R}$ be a positive monotone increasing function such that $f(t+1)\geq \lambda f(t)$ for some fixed $\lambda$. Consider the product $X=X_0\times \mathbb{R}$ and suppose there exists a metric $D$ on $X$ such that the embedding $t\mapsto (x_0,t)$ is an isometry for every $x_0\in X_0$ and

\begin{align*} D[(x_1,t);(x_2,t)]\leq f(t)d(x_1,x_2) \end{align*}

for all $x_1,x_2\in X_0$ and $t\in \mathbb{R}$. He claims that $(X,D)$ is hyerbolic and i'm trying to figure it out why.

I'm thinking in use the four point condition of Gromov spaces. Set

\begin{align} z_1=(x_1,r);z_2=(x_2,s);z_3=(x_3,t);z_4=(x_4,u) \end{align}

Using the fact $\mathbb{R}$ is $0$-hyperbolic, and the assumptions over the metric i manage to get

\begin{align} D(z_1,z_2)+D(z_3,z_4)\leq 4f(T)\xi+\max\{D(z_1,z_3)+D(z_2,z_4),D(z_1,z_4)+D(z_2,z_3)\} \end{align}

where $T=\max\{r,s,t,u\}$ and $\xi=\max_{i,j}d(x_i,x_j)$. So we need to control the first therm of the above inequality. Until here, i haven't use the following deductions that could be useful, \begin{align} f(t)\longrightarrow \infty, t\rightarrow \infty;\hspace{0.5cm} f(t)\rightarrow 0, t\rightarrow -\infty \end{align}

Any suggestions? Thanks in advance.

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