I have $z \sim N(0,n)$
By my script, a random variable $\xi \sim N(0,1)$ satisfies the following tail bound for all $t \ge 0$, $$P(\xi \ge 0) \le e^{-\frac{t^2}{2}}$$
Goal of the derivation is to come up to : $$P(|z_i| > \sqrt{tn}) \le 2e^{-\frac{t}{2}}$$
What I tried is first to standardize $z_i$ as $\frac{z_i - 0}{n} \sim N(0,1)$.
From there, I get that for $\frac{z_i - 0}{n} \sim N(0,1)$ and every $t \ge 0$: $$P(\frac{z_i}{n} > t) \le e^{-\frac{t^2}{2}}$$ which is the same as: $$P(z_i > tn) \le e^{-\frac{t^2}{2}}$$
What I don't see is, how they manage to square root RHS of inequality under probability term and progress it to arrive to final goal.