I was solving this Functional Analysis problem, but I'm not sure I'm correct on this one, the problem is:
Let $\big(E,\|\bullet\|\big)$ be a normed vector space, and $(x_n)_n$ a sequence in $E$ such that $\sum_{n=1}^\infty x_n$ converges. Show that $$\left\|\sum_{n=1}^\infty x_n\right\| \leq \sum_{n=1}^\infty\|x_n\|$$
My solution:
We know that the following inequality holds for any $m\in\mathbb{N}$ (just using the triangular inequality):
$$\left\|\sum_{n=1}^m x_n\right\| \leq \sum_{n=1}^m \|x_n\|$$
Since the left side converges, the inequality still holds in the limit $m\rightarrow\infty$, hence we have the result.
My doubts:
The only thing I am not totally sure is my last statement, because I can't know if the right side converges, so how can I compare the two things?
But then I thought I can use the following argument:
The sequence $\left(\sum_{n=1}^m \|x_n\|\right)_m$ is clearly a monotone (non-decreasing) real sequence, so if it is bounded it converges and my result is correct, the other option is that the sequence is unbounded, in which case
$$\sum_{n=1}^\infty \|x_n\|=\infty$$
And then it makes sense to write:
$$\left\|\sum_{n=1}^\infty x_n\right\| < \infty$$
Can someone tell me if my reasoning is correct? Or if another way would be better to solve this problem?