How can I prove that the hyperbolic space $\mathbb{H}^n$ (in any of its realization as hyperboloid, Poincaré disc or Poincaré half-space) is $\delta$-hyperbolic? For the moment I am not interested in the optimal $\delta$.

Almost every source I looked upon states it as well-known, but I couldn't find a thorough proof. I tried to follow the hints on "Embeddings of Gromov hyperbolic spaces" by Bonk and Schramm, but I don't see how to connect the Bounded metric space case to the hyperbolic space. I also tried to prove the 4 points inequality explicitly for the hyperboloid in $\mathbb{R}^{n+1}$ with distance $$d(A,B)=\mathrm{arcosh}(-\langle A\mid B\rangle),\text{ where }\langle A\mid B\rangle=\sum_{i=1}^na_ib_i-a_{n+1}b_{n+1},$$ but I got stuck.

Is there any trick I can use? Or is there some easy argument that I'm missing?

Many thanks to anyone who could help me!

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