Symmetric random walk does not converge a.s. using CLT

Question

Let $(X_n)_n$ be a sequence of iid random variables with $P(X_1 = 1) = \frac{1}{2}$ and $P(X_1 = -1) = \frac{1}{2}$. Define $S_n := \sum_{i = 1}^n X_i$. Prove that $S_n$ does not converge almost surely by contradiction using the Central Limit Theorem.

The CLT states that: $\frac{S_n-nE(X_1)}{\sqrt{Var(X_1)n}} = \frac{S_n}{\sqrt{n}} \overset{\mathcal{L}}{\longrightarrow} \mathcal{N}(0,1)$.

I don't really have an idea how to deduce that $S_n$ is not almost surely convergent using the CLT.

Answer 1

Suppose $S_n\to S$ almost surely for some random variable $S$. Then $$ \frac{S_n}{\sqrt{n}}\to 0\quad a.s. $$

One the other hand, almost surely convergence implies convergence in distribution. What contradiction can you have?