I was reading [this](https://link.springer.com/book/10.1007/BFb0084913) book by Coorneart, Delzant, and Papadopoulos. I am stuck with this proposition in Chapter 1 (Proposition 1.5)



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



Bounded subsets of $\delta$-hyperbolic metric spaces
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 $\delta' = \delta + 6\eta$.


