LemmaStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-09-01

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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An absolutely convergent series has Cauchy partial sums

Statement

Let $V$ be a normed space and let $\sum_{n = 0}^{\infty} x_{n}$ be an absolutely convergent series in $V$ . Then the partial sums form a Cauchy sequence in $V$ .

Facts & Assumptions

Given: A normed space $V$ , a sequence $(x_{n})$ in $V$ , and its partial sums $s_{m} : = \sum_{n < m} x_{n}$ .

[L1]

Absolute convergence means the scalar series $\sum_{n = 0}^{\infty} ∥ x_{n} ∥$ converges (Series and absolute convergence in a normed space).

[L2]

A convergent scalar series has Cauchy partial sums (Series, partial sums, convergence and the sum, divergence, and the tail series).

Proof

technique · direct

1.1

By [L1] and [L2], for every $ε > 0$ there is $N$ such that $\sum_{ℓ \leq n < m} ∥ x_{n} ∥ < ε$ whenever $m > ℓ \geq N$ .

L1L2

1.2

For such $m > ℓ \geq N$ , the tail of the vector partial sums satisfies $s_{m} - s_{ℓ} = \sum_{ℓ \leq n < m} x_{n}$ .

given

2.1

Applying the triangle inequality to the finite sum in step 1.2 gives $∥ s_{m} - s_{ℓ} ∥ \leq \sum_{ℓ \leq n < m} ∥ x_{n} ∥ < ε$ .

step 1.1step 1.2algebra

3.1

Since this holds for every $ε > 0$ , the partial sums $(s_{m})$ are Cauchy in the norm metric, which is exactly the claim.

step 2.1given∎

Depends on

Used by

Series criterion for Banach spaces Theorem

Dependency tree · two levels

16 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

Andrew Lin and Casey Rodriguez, MIT 18.102 Introduction to Functional Analysis (standard reference, not scraped)
Gerald Teschl, Topics in Real and Functional Analysis (standard reference, not scraped)