Suppose $x_n \to L$, prove that $\frac{x_1 + \dots + x_n}{n} \to L$.
So far I've got that because $x_n \to L$, then for any $\epsilon>0$, there exists some $N \in \mathbb{N}$ such that $|x_n-L|< \epsilon$ whenever $n \geq \mathbb{N}$. From this it's not hard to show that $|x_{N}+\dots + x_m - (m-N)L| < \epsilon$, but I don't know how to deal with the beginning of the series $x_1 + \dots + x_{N-1}$, which I need to ultimately show
$ |\frac{x_1+\dots+x_{N-1}}{n} + \frac{x_N + \dots + x_n}{n} - L|<\epsilon $ whenever $n \geq K$.
