A question involving Lindeberg-Levy CLT

Question

Suppose $X_k$ are i.i.d. random variables with $E(X_1)=0$ and $0< E(X_1^2)<\infty$. Define $$S_n=\sum_{k=1}^nX_k$$ I want to show that $\liminf_{n\rightarrow \infty}S_n=-\infty$ and $\limsup_{n\rightarrow \infty}S_n=\infty$ almost surely. Now I know I am supposed to use Lindeberg-Levy CLT, which states that $$\frac{1}{\sqrt{n}}\sum_{k=1}^n X_i\xrightarrow{D}N(0,E(X_1^2))$$ in this specific case. However, I am having trouble using the convergence in distribution to show that $$P(\omega\in\Omega:\liminf_{n\rightarrow \infty}S_n=-\infty)=1$$ A hint would be appreciated.

Answer 1

Hint: the events $A = \{ \liminf_{n \to \infty} S_n = -\infty \}$ and $B = \{ \limsup_{n \to \infty} S_n =\infty\}$ are tail events, so Kolmogorov's $0-1$ law tells us that $P(A), P(B) \in \{ 0,1 \}$. Now use the CLT to show that in fact $P(A), P(B) > 0$ and conclude.