(5.4.29) Proving Integer Bounds on Rational Numbers Using Well-Ordering Principle & Archimedean Prop

In this presentation, I discuss how to establish the existence of integer bounds for any rational number by utilizing the Archimedean property and the well-ordering principle. I begin by introducing the Archimedean property, which asserts that for every rational number r, there exists an integer n such that n is greater than r. This foundational property is essential in understanding the behavior of rational numbers in relation to integers. I also note that for any rational number r, the negative of that number, -r, is also rational, which plays a critical role in our proof.

Next, I aim to prove that for any rational number r, there is an integer m such that m is less than or equal to r and r is less than m plus one. To illustrate this, I provide a concrete example using the rational number r equal to 2.7. By applying the Archimedean property, I identify an integer n that is greater than 2.7, specifically n equals 3. I then define the set S, which consists of all integers greater than 2.7, and show that 3 is indeed the least element in this set by the well-ordering principle.

From there, I assert that the integer 2 is less than or equal to 2.7 and less than 3, confirming the bounds we seek. I generalize this approach for any rational number r, again applying the Archimedean property and defining a set S of integers greater than r. By the well-ordering principle, I establish that S has a least element m, leading to the conclusion that m minus one is less than or equal to r and r is less than m. This confirms the existence of integer bounds for all rational numbers.

In conclusion, I emphasize the importance of the Archimedean property and the well-ordering principle in demonstrating the relationship between rational numbers and integers. This proof serves as a fundamental example of how mathematical principles can be applied to establish clear bounds within the number system. Thank you for your attention.

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