ExampleConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passverified 2026-08-09 (gpt-5.6-terra-codex-subscription)

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The functions $f_n(k)=1$ for $k\ge n$ and $0$ otherwise converge pointwise but not uniformly on $\mathbb{N}$

Example

Give $\{0,1\}$ the uniformity of its zero-one metric. For $n,k\in\mathbb N$ , let $f_n(k)=1$ when $k\ge n$ and $f_n(k)=0$ otherwise. Then $f_n$ converges pointwise, but not uniformly, to the zero function.

Facts & Assumptions

Given: The function set $\{0,1\}^{\mathbb N}$ .

[L1]

For $d(u,v)=0$ when $u=v$ and $d(u,v)=1$ otherwise, separation and symmetry are immediate, while the triangle inequality follows because $u\ne w$ forces $u\ne v$ or $v\ne w$ . Thus $d$ is a metric; its radius- $1/2$ balls are singletons, so its topology is discrete, and its radius- $1/2$ entourage is equality (Metric space: $d(x,y) = 0$ iff $x = y$ , symmetry, and the triangle inequality; pseudometric and ultrametric, A metric on a nonempty set generates an entourage uniformity whose induced topology and uniformly continuous maps are the usual metric notions, and this uniformity is separated, The discrete, indiscrete, cofinite, cocountable, particular-point and Sierpinski topologies).

[L2]

Pointwise and uniform convergence are the two function-space uniformities of The pointwise and uniform-convergence uniformities on a function set $Y^X$ .

[L3]

Sequence convergence means eventual membership in every neighbourhood (Convergence and cluster points of a sequence in a topological space, sequential continuity, and the sequential closure).

Verification

technique · direct

1.1

For fixed $k$ , all $n>k$ have $f_n(k)=0$ , so the coordinate sequence converges to $0$ .

L1L3

1.2

For every $n$ , $f_n(n)=1$ , so $f_n$ is not in the uniform entourage induced by equality on $\{0,1\}$ .

L1L2

2.1

Thus $f_n$ converges pointwise to zero by [L2].

step 1.1L2

3.1

Hence convergence is not uniform.

step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 87 results over 17 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.

The functions fn(k)=1f_n(k)=1fn​(k)=1 for k≥nk\ge nk≥n and 000 otherwise converge pointwise but not uniformly on N\mathbb{N}N