(b) Let = {A; : i EN} be an indexed family of sets with the property that A₁ A₂ 22 A, 2..., that is, ... C A; C... CA₂ C A₁. Find A. Justify your answer giving a rigorous proof. 20 i=1
(b) Let = {A; : i EN} be an indexed family of sets with the property that A₁ A₂ 22 A, 2..., that is, ... C A; C... CA₂ C A₁. Find A. Justify your answer giving a rigorous proof. 20 i=1
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![The problem given is as follows:
Let \(\mathcal{A} = \{ A_i : i \in \mathbb{N} \}\) be an indexed family of sets with the property that
\[ A_1 \supseteq A_2 \supseteq \cdots \supseteq A_i \supseteq \cdots, \]
that is,
\[ \cdots \subseteq A_i \subseteq \cdots \subseteq A_2 \subseteq A_1. \]
Find \(\bigcap_{i=1}^{20} A_i\). Justify your answer giving a rigorous proof.
**Explanation:**
The problem involves a family of sets \(\{A_i\}\) where each set is a subset of the previous set, forming a non-increasing chain of sets. You are required to find the intersection of the first 20 sets, \(\bigcap_{i=1}^{20} A_i\), and provide a rigorous proof of your answer.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe9b346ae-ef24-4d99-babf-e95763a360cd%2F5d5145d2-a874-4bc8-b7bb-676c5121e91b%2Fee0783q_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The problem given is as follows:
Let \(\mathcal{A} = \{ A_i : i \in \mathbb{N} \}\) be an indexed family of sets with the property that
\[ A_1 \supseteq A_2 \supseteq \cdots \supseteq A_i \supseteq \cdots, \]
that is,
\[ \cdots \subseteq A_i \subseteq \cdots \subseteq A_2 \subseteq A_1. \]
Find \(\bigcap_{i=1}^{20} A_i\). Justify your answer giving a rigorous proof.
**Explanation:**
The problem involves a family of sets \(\{A_i\}\) where each set is a subset of the previous set, forming a non-increasing chain of sets. You are required to find the intersection of the first 20 sets, \(\bigcap_{i=1}^{20} A_i\), and provide a rigorous proof of your answer.
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