= Absolute-series criterion for completeness
{title2=$X\text{ complete}\iff\sum_n\|x_n\|<\infty\Rightarrow\sum_nx_n\text{ converges}$}
A <normed vector space> is a <Banach space> exactly when every absolutely convergent <series> converges in the space. Completeness implies the assertion because the <triangle inequality> makes the partial sums a <Cauchy sequence>. Conversely, select from any <Cauchy sequence> a subsequence whose consecutive differences have <norm> at most $2^{-k}$. The <series> of these differences converges by hypothesis, so the subsequence converges. The original <Cauchy sequence> then has the same limit. This provides a useful way to demonstrate incompleteness by a summable-norm <series> whose candidate sum lies outside the space.
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