Let $\left( u _ { n } \right) _ { n \in \mathbb { N } } \in \mathbb { C } ^ { \mathbb { N } }$ and $\ell \in \mathbb { C }$. Prove that $$\left( \lim _ { n \rightarrow + \infty } \sigma _ { n } = \ell \text { and } e _ { n } = o \left( \frac { 1 } { n } \right) \right) \Rightarrow \left( \lim _ { n \rightarrow + \infty } u _ { n } = \ell \right) \qquad \text{(Weak Hardy)}$$ Hint: one may prove that for all $n \geqslant 1$, $$\sum _ { k = 0 } ^ { n } k e _ { k } = n u _ { n + 1 } - \sum _ { k = 1 } ^ { n } u _ { k }$$
Let $\left( u _ { n } \right) _ { n \in \mathbb { N } } \in \mathbb { C } ^ { \mathbb { N } }$ and $\ell \in \mathbb { C }$. Prove that
$$\left( \lim _ { n \rightarrow + \infty } \sigma _ { n } = \ell \text { and } e _ { n } = o \left( \frac { 1 } { n } \right) \right) \Rightarrow \left( \lim _ { n \rightarrow + \infty } u _ { n } = \ell \right) \qquad \text{(Weak Hardy)}$$
Hint: one may prove that for all $n \geqslant 1$,
$$\sum _ { k = 0 } ^ { n } k e _ { k } = n u _ { n + 1 } - \sum _ { k = 1 } ^ { n } u _ { k }$$