= Digit expansion in a discretely valued field
{title2=$x=\sum_{n=v(x)}^\infty a_n\pi^n$}
For a normalized <discrete valuation>, a <uniformizer> $\pi$, and a chosen set $A$ of representatives of the <residue field>, every existing nonzero element has a unique displayed expansion, with its first digit not in the maximal ideal. Starting from $y_0=\pi^{-v(x)}x$, choose $a_j$ representing $y_j$ modulo the maximal ideal and put $y_{j+1}=(y_j-a_j)/\pi$. The error after $r$ digits has valuation at least $v(x)+r$, so the sums converge to $x$ even without <completeness>. Conversely arbitrary digit sequences need not converge in an incomplete field: in $\mathbb Q$ with its three-adic valuation, the digits of $\sqrt{10}\in\mathbb Q_3$ give a counterexample. <Hensel lemma> gives this square root, but it is not rational.
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