Proof. Let $X$ be a normed space with norm $|\cdot |$ and $(x_n)$ be Cauchy. Then for all $\epsilon \gt 0, \ \exists N : m,n \gt N \implies |x_m - x_n| \lt \epsilon$ is the standard definition of Cauchy sequence. But it's easy to show that $||x_n| - |x_m|| \leq |x_m - x_n|$ and thus the sequence $|x_n|$ in $\mathbb{R}$ is Cauchy. By completeness of the reals under the absolute value norm, we have that $|x_n|$ approaches a limit and thus $(x_n)$ is absolutely convergent.
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asked Oct 3, 2013 at 4:54
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$\begingroup$ Is your normed space complete? Or does your definition of absolute convergence not entail convergence as well? $\endgroup$– Andrés E. CaicedoOct 3, 2013 at 4:55
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$\begingroup$ No, not complete. This is a lemma to showing that a normed space is a Banach space iff absolutely convergent sequences converge. $\endgroup$– Daniel DonnellyOct 3, 2013 at 4:56
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1$\begingroup$ Ah, I see. The argument you indicate is correct. $\endgroup$– Andrés E. CaicedoOct 3, 2013 at 6:04
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$\begingroup$ Do you have a reference for this lemma? $\endgroup$– Jeremy Jeffrey JamesMar 13, 2019 at 14:57
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The proof is correct. Applied more generally, it shows the following:
If $X$ and $Y$ are metric spaces, $(x_n)$ is Cauchy in $X$, and $f: X\to Y$ is a uniformly continuous map, then the sequence $f(x_n)$ has a limit.
The particular statement uses $Y=\mathbb R$ and $f(x)=\|x\|$ (which is a Lipschitz function).
That said, I don't understand the bigger picture. Apparently "absolutely convergent sequence" here means a sequence $(x_n)$ such that $\|x_n\|$ has a limit. This is the first time I see this term used anywhere (and I kind of hope it's the last one. It seems designed to confuse people.) More importantly, this notion of "absolutely convergent sequence" does not imply usual convergence, e.g., consider $x_n=(-1)^n$ in $\mathbb R$. And since $\mathbb R$ is a Banach space, this disproves the claim made in a comment, "a normed space is a Banach space iff absolutely convergent sequences converge".
Perhaps the intended claim was "a normed space is a Banach space iff absolutely convergent series converge". That is indeed correct, but then the argument given in the OP is not really relevant.