# Verifying sub-exponential property when a random variable is not sub-gaussian pro

I am referring to the Example 2.4 (page 16) in this book chapter https://www.stat.berkeley.edu/~mjwain/stat210b/Chap2_TailBounds_Jan22_2015.pdf

Suppose $$Z \sim N(0,1)$$ and random variable $$X=Z^2$$. For $$\lambda < \frac{1}{2}$$, we have $$\mathbb{E}[e^{\lambda(X-1)}] = \frac{e^{-\lambda}}{\sqrt{1-2\lambda}}$$.

In Example 2.4 it is stated that based on some calculus we can verify that

$$\frac{e^{-\lambda}}{\sqrt{1-2\lambda}} \leq e^{2\lambda^2}$$

I cant come up with a calculs to get the above inequality. How can I do it?