Consequences of the Completeness of ℝ

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.