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1.4 Archimedes’ Axiom

To aid understanding, we often draw pictures of sets. The real numbers can be thought of as the points on a straight line which extends to infinity on the right and to minus infinity on the left. The real line is a continuum; that is, the real line has no gaps. We often write $x$ for a typical point on the real line; such an $x$ may be represented by an infinite decimal expansion. The existence of such a decimal expansion is a consequence of the following.

Archimedes’ Axiom.

Between any pair of real numbers lies a rational number.

Proof. Suppose that $a<b$ are real numbers, so $b-a>0$ . If $b-a>1$ , then some integer lies between $a$ and $b$ . Otherwise, we choose $N$ to be so large that $N(b-a)>1,$ and then there exists an integer $k$ between $Na$ and $Nb$ , so the rational number $k/N$ lies between $a$ and $b$ .