Let $X_n$ be independent random variables taking values $0$ and $1$ with $P(X=1) = p_i$.
1) Show the strong law of large numbers.
2) Show the central limit theorem holds iff $\sum p_i(1-p_i) = \infty$
1) Need to show convergence of $\frac{1}{n}\sum(X_i - p_i)$. It somehow seems to be clear that the difference is between $-1$ and $1$ but I do not really see how to formally show that.