Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space. Let $(X_n)_{n\in \mathbb{N}}$ be a sequence of real random variables in $L^1(\mathbb{P})$. Each $X_n$ has a mean $c>0$.
Suppose that $(X_n)$ satisfies the strong law of large numbers, i.e., (I omit a.s. when it's clear) $$ \limsup_n |n^{-1} S_n- c|=0, $$ where $S_n = \sum_{i=1}^n X_i$ .
Is it true that $\liminf_n S_n =\infty$? This is intuitively true because $n^{-1} S_n$ concentrates around $c>0$ for large $n$, but I'm not sure how to show this.
I can show that $\limsup_n S_n = \infty$. If $\limsup S_n=k <\infty$, then $$ \frac{S_{n}}{n}\le\frac{\limsup S_{n}}{n}=\frac{k}{n}\to 0, $$ which is a contradiction.
Hint: The answer is YES. Almost surely $\frac {S_n} n >\frac c 2$ for $n$ sufficiently large. This implies $S_n >\frac {nc} 2$ for $n$ sufficiently large, so $S_n \to \infty$ with probability $1$.