I have a problem understanding the proof of the corollary of Advanced Composition
Theorem Advanced Composition:
For all $\varepsilon,\delta,\delta' \geq 0$ the class of $(\varepsilon,\delta)$-differentially private mechanisms satisfies $(\varepsilon',k\delta+\delta')$-differential privacy under $k $-fold adaptive composition for:
\begin{equation*} \varepsilon'= \sqrt{2k\ln(1/\delta')}\varepsilon+k\varepsilon(e^\varepsilon-1) \end{equation*}
Corollary
Given target privacy parameters $0<\varepsilon'<1$ and $\delta'>0$, to ensure $(\varepsilon',k\delta+\delta')$ cumulative privacy loss over $k$ mechanisms, it suffices that each mechanism is $(\varepsilon,\delta)$-differentially private, where
\begin{equation*} \varepsilon=\frac{\varepsilon'}{2\sqrt{2k \ ln(1/\delta')}} \end{equation*}
Proof:
The theorem of Advanced Composition tells us the composition will be $(\varepsilon^*,k\delta+\delta')$ for all $\delta'$, where $\varepsilon^*=\sqrt{2k \ln(1/\delta')}\cdot \varepsilon+k\varepsilon^2$. When $\varepsilon'<1$ we have that $\varepsilon^*<\varepsilon'$.
I don't see clearly the connections between the proof and the corollary. I tried using the quadratic formula. I think it, and don't see how they get the $\varepsilon$ from the corollary.
