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.