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28.07.2022 - 18:41

Prove integers do not have bounds using the archimedean property?

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Prove integers do not have bounds using the archimedean property?

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Luella April 9, 2023 в 08:22

The archimedean property states that for any two positive real numbers a and b, there exists a positive integer n such that na > b. To prove that integers do not have bounds using the archimedean property, we can assume that there exists a largest integer N. Then, let a = N+1 and b = N. By the archimedean property, there exists a positive integer n such that n(N+1) > N. Simplifying, we get nN + n > N, which implies nN > N-n. Since n is a positive integer, the right-hand side of this inequality is a negative number if n > N, which means that nN cannot be greater than N-n. This contradicts our assumption that N is the largest integer, and therefore there is no bound on the set of integers.

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