Let $X_1,X_2,...$ be a sequence of i.i.d. random variables that take the value $1$ with probability $0.5$ and the value $-1$ with probability $0.5$, and let $Z$ be a standard normal random variable. Find a constant $C$ such that:
$\left |P \left (\frac{1}{\sqrt{n}} \displaystyle \sum_{i=1}^nX_i \leq t \right)-P(Z \leq t) \right | \leq \frac{C}{\sqrt n}$
I know you can use something like Berry-Essen, but I was wondering if there is a more elementary method if we don't care about how strict the bound $C$ is. I would really appreciate any help. Thanks!