Prove that, if $\lim_{j\rightarrow\infty}a_j= \ell$, then $\lim_{j\rightarrow\infty}m_j= \ell$.
How to prove that if $\lim_{n \rightarrow \infty}a_n=A$, then $\lim_{n \rightarrow \infty}\frac{a_1+...+a_n}{n}=A$