CorollaryStatement: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)precheck 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 $f_k:X\to\mathbb{R}$ . The function series $\sum f_k$ converges uniformly on $X$ if and only if, for every real $\varepsilon>0$ , there is $N\in\mathbb{N}$ such that

$\left|\sum_{k=m+1}^{n}f_k(x)\right|<\varepsilon$

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

Facts & Assumptions

Given: A set $X$ , functions $f_k:X\to\mathbb{R}$ , 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 $\varepsilon>0$ be real.

1.2

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

given

2.1

By [L1] and [L2], choose $K$ such that $|S_p(x)-S_q(x)|<\varepsilon$ for every $p,q\ge K$ and every $x\in X$ .

step 1.1L1L2choose

2.2

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

step 1.2L3algebra

3.1

For $n>m\ge K$ and $x\in X$ , the indices $n+1,m+1$ are at least $K$ , so [L3] and step 2.1 give $\left|\sum_{k=m+1}^{n}f_k(x)\right|<\varepsilon$ .

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 · next 3 levels

Direct dependencies and their dependencies through the next three levels: 29 results over 9 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

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