How do I integrate the integral for the independent Poisson random variable? I don't know how to derive this (my red pen mark)
Possible explanation:
Since $\left| -\frac{1}{\sqrt n} \right | \approx 0,$ the interval of integration is very close to zero, and so $x^2 \approx 0 \implies e^{-x^2/2} \approx 1$.
When $x \approx 0$, we have $e^{-x^2/2} \approx 1$. Since the text states that we are doing a slightly less than rigorous proof, replace $e^{x^2/2}$ with $1$ and the answer follows immediately.