Let $x_1, x_2, \ldots$ be a Cauchy sequence in a normed vector space $X$. Suppose that any absolutely convergent series converges. Then: prove that $x_1, x_2, \ldots$ converges.
My idea was: set $y_n = x_n - x_{n-1}$, then if we can prove that the sum of the $y_n$ converges absolutely, this means that $$ \sum_{n = 1}^{\infty} y_n = \lim_{k \to \infty} \sum_{n = 1}^{k} y_n = \lim_{k \to \infty} x_k $$ exists. But the problem is: how do I prove that $$ \sum_{n = 1}^{\infty} ||y_n|| $$ converges? Or is this the wrong approach? (In the latter case: don't spoil too much, please.)