(4) Prove that there is no smallest positive real number.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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Transcribed Image Text:**Problem Statement:**
(4) Prove that there is no smallest positive real number.
**Explanation:**
This statement is asking us to demonstrate why it is impossible to identify a smallest positive real number. The proof involves understanding the properties of real numbers and the concept of limits. Here's a common approach for the proof:
**Proof:**
1. **Assumption**: Assume, for the sake of contradiction, that there is a smallest positive real number, denoted by \( x \).
2. **Contradiction**: Consider the number \( \frac{x}{2} \). This number is positive and clearly less than \( x \) because \( \frac{x}{2} < x \).
3. **Conclusion**: Our assumption that \( x \) is the smallest positive real number leads to a contradiction since we've found a smaller positive real number \( \frac{x}{2} \). Therefore, no smallest positive real number exists.
This example illustrates the density property of real numbers: between any two real numbers, another real number exists. Thus, no smallest positive real number can exist.
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