Suppose that $\lbrace X_i:i=1,2,...\rbrace$ is i.i.d. with density function $f$, finite $\sigma$ and mean $\mu <0$. Prove or disprove that $$\mathbb{P}(\sum_{i=1}^n X_i \geq 0) \to 0 $$ as $n$ tends to infinity.
Note that $$\mathbb{P}(\sum_{i=1}^n X_i \geq 0) = \mathbb{P}(\frac{\bar{X}_n-\mu}{\sigma/\sqrt{n}}\geq \frac{-\mu}{\sigma/\sqrt{n}})$$ My idea is to using CLT here, but don't know how to give a $\epsilon-\delta$ argument for the convergence.
CLT is not the right approach. Use SLLN instead. Let $T_n =\frac 1 n \sum\limits_{k=1}^{n} X_k$. Then $X_n \to \mu $ almost surely. This implies convergence in distribution and hence $\lim \sup P(T_n \geq 0) \leq P(\mu \geq 0)=0$.