 Digit expansion in a discretely valued field

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.