Prove that every fixed real number is between two integers.

by RoRi

June 30, 2015

Prove that if $x \in \mathbb{R}$ is fixed, then there exist $m, n \in \mathbb{Z}$ such that $m < x < n$ .

Proof. Since the set of positive integers is unbounded above (Example #1, p. 24 of Apostol) we know there exists an $n \in \mathbb{Z}^+$ such that $x < n$ (otherwise $x$ would be an upper bound on $\mathbb{Z}^+$ ).
Then, by the same logic there is some $(-m) \in \mathbb{Z}^+$ such that $-x < -m$ ; hence, $m < x$ . Since $(-m) \in \mathbb{Z}^+$ , we know $-m \in \mathbb{Z}$ . Thus, we have found $m,n \in \mathbb{Z}$ such that $m < x < n. \qquad \blacksquare$

Stumbling Robot

Prove that every fixed real number is between two integers.

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