Let $\{X_n\}$ be i.i.d random variables. $E[X_1]=0$. Then $\sum_{i=1}^{n}X_i\over n$ converges almost surely to zero.
I know that when the sequence $\{X_n\}$ satisfies $\sum_{i=1}^{\infty}{Var(X_i)\over i^2} \lt \infty$, the conclusion holds. So I tried to cut $X_n$ to make it satisfy the condition.
Let $Y_n=X_n1_{\{a_n\le X_n\lt b_n\}}$, choose proper $a_n$ and $b_n$ can let $Y_n$ satisfy the condition. I also hope to let $E[Y_n]=0$, but that is hard.
Any advice on this problem will be appreciated.
Let $N \in \mathbb N$ and write $$X_n=X_n1_{\{|X_n|\le N\}}+X_n1_{\{|X_n|>N\}}$$ or in your notation (which however confuses $N$ with $n$) $$X_n=Y_n+X_n1_{\{|X_n|>N\}}$$ So, actually $b_n=N=-a_n$ or to avoid confusion $b_N=N=-a_N$. Now, showing that $EY_n=0$ is not possible, but $$EY_n\to EX_n=0$$ as $N\to +\infty$.