TAOCP 1.2.2 Exercise 1

There is no smallest positive rational number.

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Section 1.2.2: Numbers, Powers, and Logarithms

Exercise 1. [00] What is the smallest positive rational number?

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Solve time: 32s

There is no smallest positive rational number. Suppose, for contradiction, that $r > 0$ is the smallest positive rational. Then $r/2$ is also a positive rational number, and $0 < r/2 < r$, contradicting the minimality of $r$. Therefore, for any positive rational number, there exists a smaller positive rational number. This completes the proof.

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