Suppose two sequence $(a_n)$ and $(b_n)$ where $$b_n := \frac{a_1 + ...+ a_n}{n}$$ I want to show that if $a_n \rightarrow a$, then $b_n \rightarrow a$.
Hence, for all $\epsilon > 0$, there exists $N \in \mathbb{N}$ such that if $n \geq N$, $n \in \mathbb{N}$, then $|a_n - a| < \epsilon$.
$$\left|\frac{a_1 + ...+ a_n}{n} - \frac{an}{n}\right| = \left|\frac{(a_1 -a) + ...+ (a_n - a)}{n}\right| \leq \frac{1}{n}(|a_1 - a| + ... + |a_n - a|)$$ I know I can use $|a_n - a| < \epsilon$, but I don't know what will happen with $a_1, ..., a_{n-1}$. Do I have to use induction here? Can you give me a hint?
