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(Exercise 1.1.9) Let A, B C R be non-empty bounded sets such that B C A. Suppose that for all ï ¤ A, there exists a y € B such that x ≥ y. Show that inf B = inf A.

(Exercise 1.1.9) Let A, B C R be non-empty bounded sets such that B C A. Suppose that for all ï ¤ A, there exists a y € B such that x ≥ y. Show that inf B = inf A.

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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Analysis

Review some useful properties of sup/inf.

**Exercise 1.1.9** Let \( A, B \subset \mathbb{R} \) be non-empty bounded sets such that \( B \subset A \). Suppose that for all \( x \in A \), there exists a \( y \in B \) such that \( x \geq y \). Show that \( \inf B = \inf A \). **Hint:** You may find the following variant of Proposition 1.2.8 helpful: If \( S \subset \mathbb{R} \) is a nonempty bounded below set, then for every \( \varepsilon > 0 \) there exists \( x \in S \) such that \(\inf S \leq x < \inf S + \varepsilon\).
Transcribed Image Text:**Exercise 1.1.9** Let \( A, B \subset \mathbb{R} \) be non-empty bounded sets such that \( B \subset A \). Suppose that for all \( x \in A \), there exists a \( y \in B \) such that \( x \geq y \). Show that \( \inf B = \inf A \). **Hint:** You may find the following variant of Proposition 1.2.8 helpful: If \( S \subset \mathbb{R} \) is a nonempty bounded below set, then for every \( \varepsilon > 0 \) there exists \( x \in S \) such that \(\inf S \leq x < \inf S + \varepsilon\).
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