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How to prove that there’s always at least one rational number between two real numbers?

1 Answer

Feb 3, 2018

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Explanation:

Given two real numbers $a < b$ $a < b$ , choose an integer $N > \frac{1}{b - a}$ $N > 1/(b-a)$

Then $b - a > \frac{1}{N}$ $b-a > 1/N$

If $b$ $b$ is an integer multiple of $\frac{1}{N}$ $1/N$ then $b - \frac{1}{N}$ $b-1/N$ is a rational number in $(a, b)$ $(a, b)$

Otherwise $\frac{⌊ b N ⌋}{N}$ $floor(bN)/N$ is a rational number in $(a, b)$ $(a, b)$

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