[Math] Show that if $\sum x_n$ converges then $x_n \to 0$

Question

Let $(V,\|\|)$be a normed space. Let $(x_n) \subset V^{\Bbb{N}}$. We say that $\sum x_n$ converges if, $\lim_{n\to \infty} \sum_{i=1}^{n}x_i$ exists.

Show that if $\sum x_n$ converges then $x_n \to 0$

Consider $s_n = \sum_{i=1}^{n}x_i$. Then since $\sum x_n$ converges we have that $s_n \to l$ for some $l \in V$. So I wan't to see that given $\epsilon >0 $ there exist $n_0$ such that $\|x_n\| < \epsilon$ if $n \geq n_0$. So I'm trying to use the fact that $(s_n)$ is a Cauchy sequence, but im only managing to get an expression like

$$\|s_{n+p}-s_n\|=\|x_{n+p}+…+x_{n+1}\|< \epsilon$$

If $n$ is sufficiently big. But I don't know how to bound my $\|x_n\|$ from that expression. Any hints?

Answer 1

Hint

Try letting $p=1$.

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