CorollaryStatement: AI-adaptedProof: AI-generatedprecheck passverified 2026-08-10 (gpt-5.6-terra-codex-subscription)

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A series of real-valued functions converges uniformly if and only if its tails are uniformly small

Statement

Let $X$ be a set and let $f_{k} : X \to R$ . The function series $\sum f_{k}$ converges uniformly on $X$ if and only if, for every real $ε > 0$ , there is $N \in N$ such that

$∣ \sum_{k = m + 1}^{n} f_{k} (x) ∣ < ε$

for every $n > m \geq N$ and every $x \in X$ .

Facts & Assumptions

Given: A set $X$ , functions $f_{k} : X \to R$ , and partial-sum functions $S_{r} (x) = \sum_{k < r} f_{k} (x)$ .

[L1]

The series $\sum f_{k}$ converges uniformly exactly when its partial-sum sequence $(S_{r})$ converges uniformly (A series of real-valued functions and its pointwise and uniform convergence through its partial sums).

[L2]

A sequence of real-valued functions converges uniformly exactly when it is uniformly Cauchy (A sequence of real-valued functions converges uniformly if and only if it is uniformly Cauchy).

[L3]

For $n > m$ , $\sum_{k = m + 1}^{n} f_{k} (x) = S_{n + 1} (x) - S_{m + 1} (x)$ , and $∣ - u ∣ = ∣ u ∣$ for every real $u$ (A series of real-valued functions and its pointwise and uniform convergence through its partial sums, Basic properties of the absolute value).

Proof

technique · direct

1.1

Suppose first that $\sum f_{k}$ converges uniformly, and let $ε > 0$ be real.

1.2

Conversely, suppose the displayed tail condition holds, and fix a real $ε > 0$ and a corresponding index $N$ .

given

2.1

By [L1] and [L2], choose $K$ such that $∣ S_{p} (x) - S_{q} (x) ∣ < ε$ for every $p, q \geq K$ and every $x \in X$ .

step 1.1L1L2choose

2.2

Put $K : = N + 1$ . If $p, q \geq K$ and $x \in X$ , then the difference is $0$ when $p = q$ ; if $p > q$ , set $m = q - 1 \geq N$ and $n = p - 1 > m$ , so the tail condition and [L3] give $∣ S_{p} (x) - S_{q} (x) ∣ < ε$ ; the case $q > p$ follows by symmetry of absolute value.

step 1.2L3algebra

3.1

For $n > m \geq K$ and $x \in X$ , the indices $n + 1, m + 1$ are at least $K$ , so [L3] and step 2.1 give $∣ \sum_{k = m + 1}^{n} f_{k} (x) ∣ < ε$ .

step 2.1L3

3.2

Thus $(S_{r})$ is uniformly Cauchy, hence converges uniformly by [L2], and therefore $\sum f_{k}$ converges uniformly by [L1].

step 2.2L1L2

4.1

Steps 3.1 and 3.2 prove the two implications, hence the equivalence.

step 3.1step 3.2∎

Depends on

Used by

Dependency tree · two levels

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Sources

Mathematics LibreTexts, Sequences and Series of Functions (standard reference, not scraped)
W. Trench, Introduction to Real Analysis (standard reference, not scraped)