As stated, if $\{a\}\_{n=1}^\infty$ converges to $a \in \mathbb{R}$, prove $$\lim\_{n \to\infty} \sigma\_{N} = \lim\_{N \to\infty}\frac{1}{N} \sum\_{n = 1}^{N} a\_n = a$$

Not sure how to begin with this. I know $\sum a\_n$ won't converge unless $a=0$. It also seems to me that $\lim\_{n \to\infty} \sigma\_{N} = 0 + 0 + ... + 0$ since $a\_n \in \mathbb{R}$, which I know can't be right.

Thank you!