Let us in a normed linear space have a sequence $\{a_i\}_{i=1}^\infty$ which converges to some value $b$, how can I show that $\lim_{n\rightarrow\infty}\sum_{i=1}^n\frac{a_i}{n}=b$ My idea is to use a theorem which states that since the series is convergent, for each $\varepsilon>0$ there exist $n\in\mathbf{N}$ such that $d(x_j,x_k) < \varepsilon$ if $k > n$ and $j > n$. Then I think of splitting the sum in two parts and at the same time let $n\rightarrow\infty$ and $\varepsilon\rightarrow0$. The first part of the sum should then tend to $0$ while the second part should thend to $b$.
Does this idea hold? Any better ideas?