The proof of lower bound on regret of a two-armed bandit introduces this magical event.

\begin{equation} C_n = \{ T_2(n) > \frac{1- \epsilon}{kl(\mu_2, \mu_2^\prime)} \ln n, \hat{kl}_{T_2(n)} \leq (1 - \frac{\epsilon}{2} ln(n))\}, \end{equation}

where $T_2(n)$ in the number of time arm 2 was played until time $n$, $\epsilon > 0$, $\hat{kl}_s$ is the empirical KL-divergence of $kl(\mu_2, \mu_2^\prime)$ and $kl(p, q) = p \ln \frac{p}{q} + (1-p) \ln \frac{1-p}{1-q}$. I have read this (or similar) proof in a number of papers but none of the papers tell how they came up with the event $C_n$. Could someone tell how was this event thought of?