Let $\left( u _ { n } \right) _ { n \in \mathbb { N } } \in \mathbb { C } ^ { \mathbb { N } }$ and $\ell \in \mathbb { C }$. Prove that
$$\left( \lim _ { n \rightarrow + \infty } u _ { n } = \ell \right) \Rightarrow \left( \lim _ { n \rightarrow + \infty } \sigma _ { n } = \ell \right)$$
where $\sigma_n = \frac{1}{n+1}\sum_{k=0}^n u_k$.
If $\left( u _ { n } \right) _ { n \in \mathbb { N } }$ takes real values, prove that the result holds if $\ell = + \infty$ or $\ell = - \infty$.