Let $R \in \mathbb{R}^{n,d}$ be a random Gaussian matrix comprised of independent $\operatorname{N}(0,\frac{1}{n})$ entries, let $\textbf{w}$ and $\textbf{x}$ be vectors in $\mathbb{R}^d$, and let $\epsilon \in (0,1)$. In Shi et al. 2012, it is claimed in Lemma 10 that $$\operatorname{Pr}(1-\epsilon \leq \frac{\|R\textbf{x}\|^2}{\|\textbf{x}\|^2} \leq 1 + \epsilon) \geq 1 - 2\exp(-\frac{n}{2}(\frac{\epsilon^2}{2}-\frac{\epsilon^3}{3}))$$

I don't understand how this lower bound was obtained. It seems to me that when you want to find a lower bound on a probability in a situation like this, you should try to minimize $$\operatorname{Pr}\bigg(\frac{\|R\textbf{x}\|^2}{\|\textbf{x}\|^2} \leq 1 + \epsilon \bigg)$$ and maximize $$\operatorname{Pr}\bigg(\frac{\|R\textbf{x}\|^2}{\|\textbf{x}\|^2} \leq 1 - \epsilon \bigg)$$

If that's what Shi et al. did in this situation, I don't understand how they did it.