Let $_1,_2, … , _$ be a sequence of independent random variables with $(_ = 4^) = (_ = −4^) = \frac12$. Let $_ = _1 + _2 + ⋯ + _$. Determine for which $ > 0$, if any, that the $P(\frac{1}{}|S_n|≥ )$ does not converge to $0$. Why does this result not contradict with the weak law of large numbers?
To determine for which $ > 0$, I tried to use chebyshev's inequality, $P(\frac{1}{}|S_n|≥ )≤\frac {Var(S_n)}{^2} = \frac {Var(X_1+X_2+...+X_n)}{^2} = \frac{Var(X_1)+Var(X_2)+...+Var(X_n)}{^2}$ since $_1,_2, … , _$ are independent, then I took $0≤\frac{Var(X_1)+Var(X_2)+...+Var(X_n)}{^2}≤1$ and find the range of $\epsilon$ , is that correct? and how to relate the result with WLLN?