$\{X_n\}$ are i.i.d and nonnegative r.v.s such that $\mathbb{E}X_1=+\infty$, prove that $\frac{1}{n}\sum_{i=1}^n X_i \to +\infty$, a.s.

Question

I have $\mathbb{P}(\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^n X_i=+\infty)=0 \ \mbox{or} \ 1$, according to the Kolmogorov 0-1 law.

I want to use the method of contradiction: Suppose $\frac{1}{n}\sum_{i=1}^n X_i$ does not converge to $+\infty$, a.s., i.e. $\mathbb{P}(\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^n X_i=+\infty)=0$. In other words, $\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^n X_i<+\infty$, a.s. And, I want to get contradiction by \begin{equation} \infty>\mathbb{E}(\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^n X_i)=\lim_{n\to\infty}\mathbb{E}(\frac{1}{n}\sum_{i=1}^n X_i)=\mathbb{E}X_1=+\infty \end{equation} But there are two problems,

1. $X<\infty$, a.s. can't guarantee that $\mathbb{E}X<\infty$
2. I can’t use the DCT nor the MCT to exchange the order of limits.

I wonder know if $\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^n X_i<+\infty$, a.s. is equivalent to $\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^n X_i\to $ constant, a.s.？Because $\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^n X_i$ is measurable with respect to tail event, maybe a constant. Please tell me if this is right? If yes, two problems above are solved.