 Solution

A necessary qualification is ${\sigma }^2>0$. As printed, the assumptions also permit $X_i=0$ almost surely; then $\mathbb{P}(A_n)=0$ for every $K>0$, so the asserted positive lower bound is false. Assume the intended nondegenerate case.

The central limit theorem states that independent identically distributed variables with mean zero and finite positive variance satisfy $S_n/(\sigma \sqrt{n})\Rightarrow Z$, where $Z\sim N(0,1)$. Since the normal distribution has no atom at $K/\sigma$,For example,