I was particularly interested in the following:
When I read this proof, everything seemed fine and logical except one detail (the proof is located here).
Right after we prove, that the series $\sum_{k=1}^\infty x_{N_m} - x_{N_m+1}$ converges, there is a statement which tells us that the limit of that series (let's name it $s$) definitely belongs to the initial space $\mathbb{X}$: $s \in \mathbb{X}$.
Why is that? That could probably be very obvious, but unfortunately I can't get it.
Why should a $lim \hspace{2 mm} \sum_{k=1}^\infty (x_{i} - x_{j})$, where $x_i \in \mathbb{X}$ belong to $\mathbb{X}$ itself?
What am I missing? :)