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If E≤R, say that x is a limit point of E if for every € > 0, there is a point e in E with 0 < xe|< €. (Note that we insist that ex though there is nothing to preclude x being in the set E.) A point x is called an isolated point of E if x & E but x is not a limit point. (a) Show that E is a closed set if and only if it contains all its limit points. (b) Show that x is a limit point of E if and only if there is a sequence of distinct points in E that converges to x.

If E≤R, say that x is a limit point of E if for every € > 0, there is a point e in E with 0 < xe|< €. (Note that we insist that ex though there is nothing to preclude x being in the set E.) A point x is called an isolated point of E if x & E but x is not a limit point. (a) Show that E is a closed set if and only if it contains all its limit points. (b) Show that x is a limit point of E if and only if there is a sequence of distinct points in E that converges to x.

Advanced Engineering Mathematics
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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If \( E \subseteq \mathbb{R} \), say that \( x \) is a *limit point* of \( E \) if for every \( \epsilon > 0 \), there is a point \( e \) in \( E \) with \( 0 < |x - e| < \epsilon \). (Note that we insist that \( e \neq x \) though there is nothing to preclude \( x \) being in the set \( E \).) A point \( x \) is called an *isolated point* of \( E \) if \( x \in E \) but \( x \) is not a limit point. (a) Show that \( E \) is a closed set if and only if it contains all its limit points. (b) Show that \( x \) is a limit point of \( E \) if and only if there is a sequence of distinct points in \( E \) that converges to \( x \).
Transcribed Image Text:If \( E \subseteq \mathbb{R} \), say that \( x \) is a *limit point* of \( E \) if for every \( \epsilon > 0 \), there is a point \( e \) in \( E \) with \( 0 < |x - e| < \epsilon \). (Note that we insist that \( e \neq x \) though there is nothing to preclude \( x \) being in the set \( E \).) A point \( x \) is called an *isolated point* of \( E \) if \( x \in E \) but \( x \) is not a limit point. (a) Show that \( E \) is a closed set if and only if it contains all its limit points. (b) Show that \( x \) is a limit point of \( E \) if and only if there is a sequence of distinct points in \( E \) that converges to \( x \).
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