I am trying to prove that if $X$ is a Banach space, then every absolutely convergent series in $X$ converges in $X$. My current proof is below but I realize that, in the first paragraph, the first "convergent" is different than the second "convergent". How does this impact my proof, and if it impacts it negatively, how can I remedy it.
We want to prove (Cauchy$\implies$Convergent)$\implies$(Abs. Conv.$\implies$Convergent). This is equivalent to proving Abs. Conv.$\implies$Cauchy.
Suppose $\{x_n\}_\mathbf{N}$ is a sequence such that the series $\sum\|x_n\|$ is absolutely convergent. That is, $\sum\|x_n\|<\infty$, i.e., $\{\sum_{i=1}^n\}_{n\in\mathbf{N}}$ is convergent. Since, all elements of this sequence are in $\mathbf{R}$, this sequence is Cauchy. Then, by definition of Cauchy Sequences,
$\forall\epsilon>0,\exists N\in\mathbf{N}\ni\forall n,m\ge N:m\ge n,$
$\|x_m-x_n\|\le |\|x_m\|-\|x_n\|\le |\|x_m\|+\|x_n\||\le \left|\sum_{i=n}^m\|x_i\|\right|<\epsilon.$
That is, $\{x_n\}_{n\in\mathbf{N}}$ is Cauchy. Therefore, $X$ is complete implies that all absolutely convergent series in $X$ converges in $X$.