So the question I'm answering is "Suppose (X, || ||) is a normed space. Show that X is complete iff every absolutely convergent series in X converges on an element of X."
The first half was simple (show that every absolutely convergent series can be expressed as a Cauchy sequence of partial sums, and thus converges because X is complete), but the second half I'm having trouble with.
So, in my quest to show that "Every absolutely convergent series in X converges to an element in X implies X is complete," I'm trying to find a way to show that, given a Cauchy sequence $(a_n)_{n \epsilon \Bbb N}$, the series $\Sigma_{k=1}^\infty x_k$, where $x_k := a_k-a_{k-1}$ (with $a_0 = 0$) is absolutely convergent.
So far I have:
For given $\varepsilon > 0, \exists N \epsilon \Bbb N$ such that $m, n \epsilon \Bbb N, m>n>N \Rightarrow\\ ||a_m - a_n|| = ||\Sigma_{k=1}^m x_k -\Sigma_{k=1}^nx_k|| = ||\Sigma_{k=n+1}^mx_k||<\varepsilon\\$ and $||a_m-a_n||\ge|||a_m||-||a_n|||=|||\Sigma_{k=1}^m x_k|| -||\Sigma_{k=1}^nx_k|||$
and now I'm lost.
