Let $x \in \ell^p$, where $1 \leq p < \infty$.
Notice that we can write $x= \sum_{n=1}^{\infty} x_n e_n$, where $e_n=(0, \cdots, 1,0, \cdots )$ i.e. $1$ is at the nth term.
I want to show that that the rearrangement of the series $ \sum_{n\in \mathbb{N}} x_n e_n =\sum_{n=1}^{\infty} x_n e_n= x$.
This is what I have so far:
What we want to show is equivalent to showing $ \sum_{n=1}^{\infty} |x_n e_n|=x$.
$ \sum_{n=1}^{\infty} |x_n e_n|= \sum_{n=1}^{\infty} |x_n e_n|= \sum_{n=1}^{\infty} |x_n|= \| x\|_1 < \infty $
Let $\epsilon >0$.
Since $x= \sum_{n=1}^{\infty} x_n e_n$. There exists some $N$ such that for all $k>N$,
$\| x- \sum_{n=1}^{k}x_n e_n\| < \epsilon$.
So $\| x- \sum_{n=1}^{k} | x_n e_n| \|\leq \| x- \sum_{n=1}^{k}x_n e_n\|<\epsilon$.
Thus, $x= \sum_{n=1}^{\infty} |x_n e_n|$.
So we've shown that $ \sum_{n\in \mathbb{N}} x_n e_n =\sum_{n=1}^{\infty} x_n e_n= x$.
Could someone check if my steps are valid?
I'm not so sure about $x= \sum_{n=1}^{\infty} |x_n e_n|=\sum_{n\in \mathbb{N}} x_n e_n =\sum_{n=1}^{\infty} x_n e_n $
Thank you in advance!