2.3.12Please include reasons for each step. From The Real Numbers and Real Analysishttps://muhammadalfaridzi.files.wordpress.com/2014/05/the_real_numbers_and_real_analysis.pdf ### Section 2.3.12**Task**: Prove Lemma 2.3.10 (2). Only need to prove #2 of Lemma 2.3.10.---### Lemma 2.3.10Let \( a \in \mathbb{R} \).1. \( a \leq 0 \) if and only if \( a < \varepsilon \) for all \( \varepsilon > 0 \)2. \( a \geq 0 \) if and only if \( a > -\varepsilon \) for all \( \varepsilon > 0 \)3. \( a = 0 \) if and only if \( |a| < \varepsilon \) for all \( \varepsilon > 0 \)---This lemma involves proving inequalities for real numbers using limits and comparisons with small positive numbers \( \varepsilon \). The goal is to demonstrate the truth of these inequalities under the specified conditions.

Answered: Image /qna-images/answer/83b74757-8a50-43d2-9b6d-de134dcc7319.jpg

2.3.12 Prove Lemma 2.3.10 (2). Only need to prove #2 0f Lemma 2.3.10. Lemma 2.3.10. Let a E R. 1. a <0 if and only if a < ɛ for all ɛ > 0 2. a 2 0 if and omly if a > -e for all ɛ > 0 3. a = 0 if and omly if Ja| < ɛ for all ɛ > 0.

2.3.12 Prove Lemma 2.3.10 (2). Only need to prove #2 0f Lemma 2.3.10. Lemma 2.3.10. Let a E R. 1. a <0 if and only if a < ɛ for all ɛ > 0 2. a 2 0 if and omly if a > -e for all ɛ > 0 3. a = 0 if and omly if Ja| < ɛ for all ɛ > 0.

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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Question

2.3.12

Please include reasons for each step.

From The Real Numbers and Real Analysis

https://muhammadalfaridzi.files.wordpress.com/2014/05/the_real_numbers_and_real_analysis.pdf

### Section 2.3.12

**Task**: Prove Lemma 2.3.10 (2). Only need to prove #2 of Lemma 2.3.10.

---

### Lemma 2.3.10

Let \( a \in \mathbb{R} \).

1. \( a \leq 0 \) if and only if \( a < \varepsilon \) for all \( \varepsilon > 0 \)

2. \( a \geq 0 \) if and only if \( a > -\varepsilon \) for all \( \varepsilon > 0 \)

3. \( a = 0 \) if and only if \( |a| < \varepsilon \) for all \( \varepsilon > 0 \)

---

This lemma involves proving inequalities for real numbers using limits and comparisons with small positive numbers \( \varepsilon \). The goal is to demonstrate the truth of these inequalities under the specified conditions.

Branch of mathematical analysis that studies real numbers, sequences, and series of real numbers and real functions. The concepts of real analysis underpin calculus and its application to it. It also includes limits, convergence, continuity, and measure theory.

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