Bounded subsets of $\delta$-hyperbolic metric spaces

I was reading this book by Coorneart, Delzant, and Papadopoulos. I am stuck with this proposition in Chapter 1 (Proposition 1.5)

If $$Y$$ is a bounded $$\delta$$-hyperbolic subset of $$X$$, then $$X$$ is $$\delta'$$-hyperbolic with $$\delta' = \delta + 6 \eta$$ where $$\eta = \sup_{x \in X} \operatorname{dist}(x,Y)$$.

The authors have not given proof of the proposition. I have tried using the $$\delta$$-hyperbolicity condition of $$Y$$ but I am getting stuck. Please help.

• @AmirSagiv Thanks for editing the question Sep 23 '20 at 7:04

Let $$x,y,z,b \in X$$ be given and assume that $$\eta = \sup_{x\in X} d(x,Y) < \infty$$. Fix $$x' , y' , z' , b' \in Y$$ such that $$|x-x'|, |y-y'|$$, etc. are all $$\leq \eta$$. (For a 'properly done' proof, use $$|x-x'|<\eta + \varepsilon$$ for some $$\varepsilon>0$$ and take $$\varepsilon \to 0$$ at the end).
Using the triangle inequality, one can show that \begin{align*} & (x'|y')_{b'} \leq (x|y)_b + 3\eta \\ & (x|z)_b \leq (x'|z')_{b'} + 3\eta \\ \text{and } & (y|z)_b \leq (y'|z')_{b'} + 3\eta . \end{align*} Then combining the above with the fact that Y is $$\delta$$-hyperbolic, we get \begin{align*} (x|y)_b & \geq (x'|y')_{b'} -3\eta \\ & \geq \min\{ (x'|z')_{b'} , (y'|z')_{b'} \} - \delta - 3\eta \\ & \geq \min\{ (x|z)_{b} -3\eta , (y|z)_{b} -3\eta \} - \delta - 3\eta \\ & = \min\{ (x|z)_{b}, (y|z)_{b} \} - \delta - 6\eta . \end{align*} So $$X$$ is $$\delta'$$-hyperbolic with $$\delta' = \delta + 6\eta$$.