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Prove by contradiction that there are infinitely many non-negative integers.

Prove by contradiction that there are infinitely many non-negative integers.

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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**Problem 1: Infinite Non-Negative Integers** *Objective:* Prove by contradiction that there are infinitely many non-negative integers. **Explanation:** To demonstrate this idea by contradiction, we assume the opposite of what we want to prove—that there are only finitely many non-negative integers. 1. **Assumption:** Suppose there are only finitely many non-negative integers. This means the set of non-negative integers is finite. 2. **List the Integers:** Let's say the non-negative integers are listed as \(0, 1, 2, \ldots, n\) for some largest integer \(n\). 3. **Contradiction with \((n + 1)\):** However, if \(n\) is the largest non-negative integer, then \((n + 1)\) is also a non-negative integer and should be in the list. But \((n + 1)\) is greater than \(n\), which contradicts our assumption that \(n\) is the largest. 4. **Conclusion:** The assumption that there are finitely many non-negative integers leads to a contradiction. Hence, there must be infinitely many non-negative integers. This elegant proof by contradiction highlights the limitless nature of numbers and the concept of infinity in mathematics.
Transcribed Image Text:**Problem 1: Infinite Non-Negative Integers** *Objective:* Prove by contradiction that there are infinitely many non-negative integers. **Explanation:** To demonstrate this idea by contradiction, we assume the opposite of what we want to prove—that there are only finitely many non-negative integers. 1. **Assumption:** Suppose there are only finitely many non-negative integers. This means the set of non-negative integers is finite. 2. **List the Integers:** Let's say the non-negative integers are listed as \(0, 1, 2, \ldots, n\) for some largest integer \(n\). 3. **Contradiction with \((n + 1)\):** However, if \(n\) is the largest non-negative integer, then \((n + 1)\) is also a non-negative integer and should be in the list. But \((n + 1)\) is greater than \(n\), which contradicts our assumption that \(n\) is the largest. 4. **Conclusion:** The assumption that there are finitely many non-negative integers leads to a contradiction. Hence, there must be infinitely many non-negative integers. This elegant proof by contradiction highlights the limitless nature of numbers and the concept of infinity in mathematics.
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