Abstract
We have now reached the point at which we can give rigorous proofs of two facts you have known for some time. We mention the second first; that is, we will conclude this chapter by showing that there is a nonempty bounded subset of ℚ that does not have a supremum in ℚ. As an exercise, you will show that there is also a nonempty bounded subset of ℚ that does not have an infimum in ℚ. In this sense, ℚ is not complete. Since ℝ is complete, there must be numbers in ℝ that are not in ℚ. We know what these numbers are, of course; they are the irrational numbers. Thus far we haven’t proven that a particular real number is irrational. But now we can! This will be the first fact that we prove. Here’s a rough outline: We will show that if a is a positive real number, then a has a positive square root; that is, there exists \(x\, \in \mathbb{R}^ +\) such that \(x^2 \, = \,a.\) But Theorem 5.2 told us that \(\sqrt 2\) is not rational. Thus, we know \(\sqrt 2\) exists and we know it is not rational. Therefore, we have a rigorous proof that \(\sqrt 2\) is irrational.
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© 2011 Springer Science+Business Media, LLC
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Daepp, U., Gorkin, P. (2011). Consequences of the Completeness of ℝ. In: Reading, Writing, and Proving. Undergraduate Texts in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-9479-0_13
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DOI: https://doi.org/10.1007/978-1-4419-9479-0_13
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