Let $\left( u _ { n } \right) _ { n \in \mathbb { N } } \in \mathbb { R } ^ { \mathbb { N } }$ and $\ell \in \mathbb { R }$. Prove that
$$\left( \lim _ { n \rightarrow + \infty } \sigma _ { n } = \ell \text { and } \left( u _ { n } \right) _ { n \in \mathbb { N } } \text { monotone } \right) \Rightarrow \left( \lim _ { n \rightarrow + \infty } u _ { n } = \ell \right) .$$
Prove that the result holds for $\ell = + \infty$ or $\ell = - \infty$.