Strong law of large numbers

Without loss of generality assume $\mu = 0$. Truncate by setting $Y_n = X_n \mathbf{1}_{\{|X_n| \le n\}}$. Because $\sum_{n=1}^{\infty} \mathbb{P}(|X_n| > n) = \sum_{n=1}^{\infty} \mathbb{P}(|X_1| > n) \le \mathbb{E}[|X_1|] < \infty$, the Borel–Cantelli lemma implies $\mathbb{P}(X_n \ne Y_n \text{ i.o.}) = 0$. Fubini's theorem gives $\sum_{n=1}^{\infty} \frac{\operatorname{Var}(Y_n)}{n^2} \le \sum_{n=1}^{\infty} \frac{\mathbb{E}[Y_n^2]}{n^2} \le 2\mathbb{E}[|X_1|] < \infty$. By Kolmogorov's convergence criterion (derived from Kolmogorov's maximal inequality), $\sum_{n=1}^{\infty} \frac{Y_n - \mathbb{E}[Y_n]}{n}$ converges almost surely. Kronecker's lemma then yields $\frac{1}{n}\sum_{i=1}^n (Y_i - \mathbb{E}[Y_i]) \to 0$ almost surely, and since $\mathbb{E}[Y_n] \to \mu = 0$, it follows that $\bar{X}_n \to 0$ almost surely.