While reading lecture notes on Stein's method, there's one example that I cannot prove myself.
For iid random variables $X_1, ..., X_n$ where $\Pr(X_i=1) = \Pr(X_i=-1) = 1/2$, define $S_n=\frac{1}{\sqrt{n}}\sum_{i=1}^n X_i$. Then $S_n$ converges to normal distribution due to Central Limit Theorem. Still, $d_{TV}(S_n,Z)=1$ for all $n$ where $d_{TV}$ is the total variation distance and $Z\sim N(0,1)$.
It's an example of 'total variation distance is often too strong to be useful', but I don't know how to prove it. Any hint or suggestion?