I am having some trouble understanding a couple of lines in the proof of the central limit theorem using characteristic functions:
https://en.wikipedia.org/wiki/Central_limit_theorem#Proof_of_classical_CLT
I'm not sure how $(1-\frac{t^2}{2n} + o(\frac{t^2}{n}))^n \to e^\frac{-t^2}{2}$ as $n \to \infty$.
Originally I proved (by induction) a lemma that for complex numbers $w_k$ and $z_k$ if $|w_k|\leq 1$, $|z_k|\leq 1$ then $$|\prod_k{w_k}-\prod_k {z_k}| \le \sum_k{|z_k-w_k|}$$
I tried to use this to show that
$(1-\frac{t^2}{2n} + o(\frac{t^2}{n}))^n - (1-\frac{t^2}{2n})^n \to 0$ as $n\to \infty$. But I got stuck.
Instead, I've taken the logarithm of the characteristic function (see the link) and then $$n\ln[1-\frac{-t^2}{2n} + o(\frac{t^2}{n})] \stackrel{?}{=} n[\frac{-t^2}{2n} + o(\frac{t^2}{n})] = \frac{-t^2}{2} + o(1) \to \frac{-t^2}{2}$$
However, I'm not sure how to get the first equality, indicated by $?$
Any help would be greatly appreciated.
Thanks in advance.