# Series sum question

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? :)

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Because that is what it means for a series to converge in $X$. It means that the sequence of partial sums converges to an element of $X$. In general, convergence of a sequence means that there is some element of your space to which the sequence converges.
To elaborate a bit, to say that a sequence $(y_n)$ in $X$ converges without adding any qualification is another way to say that the sequence converges in $X$, which means that it has a limit in $X$. That is, there is an element $L$ of $X$ such that for all $\varepsilon\gt0$ there exists an $N$ such that $n\gt N$ implies $\|y_n-L\|\lt\varepsilon$. That is precisely the notion used in the hypothesis at your link, and it is the convergence referred to when we say that Cauchy sequences converge in complete spaces.
I always thought that convergense of a $series$ means that the partial sum sequence has a finite limit. But the definition of the convergence doesn't state that that limit should belong to the same space as the elements of the series. Or that should be obvious that it SHOULD belong to it? –  Yippie-Kai-Yay Dec 20 '10 at 19:55