generated from smyalygames/quartz
12 lines
706 B
Markdown
12 lines
706 B
Markdown
# Definition
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A **subnet** of a [[Nets|net]] $f: I \to X$ is a [[Nets|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 $h(j') \geq i \; \forall j' \geq j$.
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# Example
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> [!example]
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> ![[Drawing 2025-02-13 11.47.33.excalidraw]]
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> **Sequence:**
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> $$f : I = \mathbb{N} \to \mathbb{R} : f(n) = x_{n}$$
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> **Subsequence:**
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> $$\{ \underbrace{x_{1}}_{= x_{1}}, \, \underbrace{x_{3}}_{= x_{2}}, \, \underbrace{x_{5}}_{= x_{3}}, \, \underbrace{x_{7}}_{= x_{4}}, \, \dots \} \to 1$$
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> **Definition:**
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> $g : J = \mathbb{N} \to \mathbb{R}$ by $g(j) = x'_{j} = x_{2j-1} = f(2j-1) = f \circ h(j)$, where $h : J \to I$ is defined by $h(j) = 2j-1$.
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