= Incompleteness of finitely supported sequences
{title2=$\|x-x^{(N)}\|_1=2^{-N}$}
Equip the <vector space> $c_{00}$ of <finitely supported sequences> with the <norm> inherited from the <absolutely summable sequence space>. Consider the truncations
$$
x^{(N)}=(2^{-1},2^{-2},\ldots,2^{-N},0,0,\ldots).
$$
For $M>N$, the <geometric series> gives
$$
\|x^{(M)}-x^{(N)}\|_1=\sum_{k=N+1}^{M}2^{-k}=2^{-N}-2^{-M}<2^{-N}.
$$
Therefore $(x^{(N)})$ is a <Cauchy sequence>. Any possible <norm> limit must have coordinate $k$ equal to $2^{-k}$, since $|x_k-y_k|\leq\|x-y\|_1$. That forces infinitely many nonzero coordinates, so no limit lies in $c_{00}$. There is nothing wrong with the approximations: the space is missing their limit.
Inside the <absolutely summable sequence space>, the limit is $x=(2^{-k})_{k\geq1}$ and its exact truncation error is $2^{-N}$. Four retained coordinates leave error $1/16$; ten leave $1/1024$. More generally, truncating any element of the <absolutely summable sequence space> leaves a tail whose <norm> tends to zero. Together with <completeness>, this identifies that <Banach space> as the <completion of a normed space> $c_{00}$ with this <norm>. Finite storage and an infinite mathematical object fit together through a controlled error.
\Image[/banach-completion-tail.png]
{title=Finite-support approximations and their exact absolute-sum tail}
{description=For four retained coordinates, the omitted tail has total size $1/16$. Each extra coordinate halves the absolute-sum error, although every approximation still has finite support.}
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