Certainly! Here is the transcription of the text from the provided image:-----### Problem Statement:Let \( A_1, A_2, \ldots, A_m \) be finitely many subsets of \( \mathbb{R}^p \). Assume that \( x \) is a cluster point of \( \bigcup_{j=1}^{m} A_j \). Prove that there exists some \( 1 \leq j \leq m \) such that \( x \) is a cluster point of the single set \( A_j \).-----### Explanation:This problem is about identifying the cluster point within a collection of subsets. A cluster point of a set \( S \) in a topological space (such as \( \mathbb{R}^p \)) is a point where every neighborhood around it contains at least one point from \( S \) distinct from itself. The problem states that if a point \( x \) is a cluster point of the union of several subsets, then \( x \) must be a cluster point of at least one of those subsets individually. ### Key Concepts to Understand:1. **Cluster Point**: A point \( x \) is a cluster point of a set \( S \) if every open neighborhood of \( x \) contains at least one point of \( S \) different from \( x \) itself.2. **Union of Sets**: The union of sets \( \bigcup_{j=1}^{m} A_j \) contains all elements that belong to any of the \( A_j \) subsets.### Proof Strategy:To prove the statement, one can use:- Suppose \( x \) is indeed a cluster point of the union \( \bigcup_{j=1}^{m} A_j \).- Definition of a cluster point to argue that within every neighborhood around \( x \), there must exist a point from one of the subsets \( A_j \) which leads to the conclusion that \( x \) itself must be a cluster point of at least one \( A_j \).This is key in topology and real analysis, aiding in understanding and working with continuity and limits in higher-dimensional spaces.### Diagram or Graph Explanation:The image does not contain any graphs or diagrams to explain. Only the mathematical problem statement is present.

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Answered: Let A1, A2, ., Am be finitely many…

Let A1, A2, ., Am be finitely many subsets of R'. Assume that z is a cluster point of UA;. Prove that there exists some 1 < j < m such that z is a j=1 cluster point of the single set A,.

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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