TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-13

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$R ⟦ x ⟧$ is complete in the $x$ -adic topology and $R [x]$ is dense by truncation

Statement

Every $x$ -adically Cauchy sequence in $R ⟦ x ⟧$ has a unique $x$ -adic limit. For every $f \in R ⟦ x ⟧$ , its truncations

$f_{< N} : = \sum_{n < N} [x^{n}] f x^{n} \in R [x]$

converge $x$ -adically to $f$ . Thus $R ⟦ x ⟧$ is $x$ -adically complete and the embedded polynomial ring is dense, including when $R$ is the zero ring.

Facts & Assumptions

Given: The hypotheses and notation of the statement above.

[F1]

An $x$ -adically Cauchy sequence eventually agrees pairwise modulo every $x^{N}$ , and convergence means eventual agreement with the limit modulo every $x^{N}$ (Order of a formal series, congruence modulo $x^{N}$ , and the $x$ -adic notions of convergence and Cauchy sequence).

[F2]

The coefficientwise polynomial inclusion into formal power series is an injective unital ring homomorphism (Cauchy multiplication makes $R ⟦ x ⟧$ a commutative ring containing $R [x]$ as the finitely supported subring).

[F3]

Two formal series are equal if and only if all their coefficients are equal (Coefficient extraction is $R$ -linear, separates formal series, shifts under multiplication by $x^{k}$ , and converts products to finite convolution).

Proof

technique · stabilize coefficients

1.1

Let $(f_{j})$ be Cauchy. For each $n$ , use the Cauchy condition with $N = n + 1$ : the coefficient $[x^{n}] f_{j}$ is eventually constant. Define $[x^{n}] f$ to be that eventual value. Given $N$ , choose a common Cauchy index for the first $N$ coefficients; then $f_{j} \equiv f (m o d x^{N})$ thereafter, so $f_{j} \to f$ .

givenF1

1.2

The truncation $f_{< N}$ is finitely supported, hence belongs to the embedded $R [x]$ , and it agrees with $f$ in every degree below $N$ . Therefore $f_{< N} \to f$ ; this includes a constant or zero series and remains true in the zero ring.

givenF2

2.1

If both $f$ and $g$ are limits, then for each $N$ their first $N$ coefficients agree with the same sufficiently late $f_{j}$ . Thus every coefficient of $f$ and $g$ agrees, so $f = g$ by extensionality.

step 1.1givenF3

3.1

Existence and uniqueness are steps 1.1 and 2.1, and density is step 1.2.

step 1.1step 2.1step 1.2∎

Depends on

Used by

Rooted plane trees satisfy T(x)=x/(1-T(x)) Corollary
An order-raising recursive specification has a unique solution Theorem

Dependency tree · two levels

7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Benjamin Sambale, An Invitation to Formal Power Series (standard reference, not scraped)

R⟦x⟧R[[x]] is complete in the xx-adic topology and R[x]R[x] is dense by truncation