Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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Could you explanin how to prove 2.31 using definitions below?

**Theorem 2.30.** Let \( A \) be a subset of the topological space \( X \), and let \( p \) be a point in \( X \). If the set \(\{x_i\}_{i \in \mathbb{N}} \subset A\) and \( x_i \to p \), then \( p \) is in the closure of \( A \). As we shall see later, in some topological spaces, the converse of the previous result is not true. But it is true for \(\mathbb{R}^n\). **Theorem 2.31.** In the standard topology on \(\mathbb{R}^n\), if \( p \) is a limit point of a set \( A \), then there is a sequence of points in \( A \) that converges to \( p \). **Definition.** Let \( (X, \mathcal{J}) \) be a topological space, \( A \) a subset of \( X \), and \( p \) a point in \( X \). Then \( p \) is a *limit point* of \( A \) if and only if for each open set \( U \) containing \( p \), \( (U - \{p\}) \cap A \neq \emptyset \). Notice that \( p \) may or may not belong to \( A \). **Definition.** A *sequence* in a topological space \( X \) is a function from \( \mathbb{N} \) to \( X \). The image of \( i \) under this function is a point of \( X \) denoted \( x_i \) and we traditionally write the sequence by listing its images: \( x_1, x_2, x_3, \ldots \) or in shorter form: \((x_i)_{i \in \mathbb{N}}\). **Definition.** A point \( p \in X \) is a *limit of the sequence* \((x_i)_{i \in \mathbb{N}}\), or, equivalently, \((x_i)_{i \in \mathbb{N}}\) *converges to* \( p \) (written \( x_i \to p \)), if and only if for every open set \( U \)
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Transcribed Image Text:**Theorem 2.30.** Let \( A \) be a subset of the topological space \( X \), and let \( p \) be a point in \( X \). If the set \(\{x_i\}_{i \in \mathbb{N}} \subset A\) and \( x_i \to p \), then \( p \) is in the closure of \( A \). As we shall see later, in some topological spaces, the converse of the previous result is not true. But it is true for \(\mathbb{R}^n\). **Theorem 2.31.** In the standard topology on \(\mathbb{R}^n\), if \( p \) is a limit point of a set \( A \), then there is a sequence of points in \( A \) that converges to \( p \). **Definition.** Let \( (X, \mathcal{J}) \) be a topological space, \( A \) a subset of \( X \), and \( p \) a point in \( X \). Then \( p \) is a *limit point* of \( A \) if and only if for each open set \( U \) containing \( p \), \( (U - \{p\}) \cap A \neq \emptyset \). Notice that \( p \) may or may not belong to \( A \). **Definition.** A *sequence* in a topological space \( X \) is a function from \( \mathbb{N} \) to \( X \). The image of \( i \) under this function is a point of \( X \) denoted \( x_i \) and we traditionally write the sequence by listing its images: \( x_1, x_2, x_3, \ldots \) or in shorter form: \((x_i)_{i \in \mathbb{N}}\). **Definition.** A point \( p \in X \) is a *limit of the sequence* \((x_i)_{i \in \mathbb{N}}\), or, equivalently, \((x_i)_{i \in \mathbb{N}}\) *converges to* \( p \) (written \( x_i \to p \)), if and only if for every open set \( U \)
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