Is this true? I figured that if not, there will some positive probability $\sigma$ that $\sqrt{n}(x_n-x_0)$ takes $\sqrt{M} \cdot \epsilon$ for infinitely many large $M$. Even though this "blowing up" is weird, it does not appear to be a contradiction since we don't know how frequent of such event. That is, it could still converge to $N(0,\sigma_0^2)$ in distribution with it infinitely often taking some larger and larger values. How to write a formal proof of this(suppose it's true)?


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