Subnet

Definition

A subnet of a net $f:I\to X$ is a net $g:J\to X$ and a map $h:J\to I$ such that $g=f\circ h$ and such that $\forall i\in I$ $\exists j\in J$ with $h(j^{\prime })\ge i\forall j^{\prime }\ge j$.

Example

Example

Drawing 2025-02-13 11.47.33.excalidraw

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Sequence:
$f:I=\mathbb{N}\to \mathbb{R}:f(n)=x_n$
Subsequence:
$\{\underset{=x_1}{\underbrace{x_1}},\underset{=x_2}{\underbrace{x_3}},\underset{=x_3}{\underbrace{x_5}},\underset{=x_4}{\underbrace{x_7}},\dots \}\to 1$
Definition:
$g:J=\mathbb{N}\to \mathbb{R}$ by $g(j)=x_j^{\prime }=x_{2j-1}=f(2j-1)=f\circ h(j)$, where $h:J\to I$ is defined by $h(j)=2j-1$.