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Question

Prove that between any two points a and b ($a\not=b$) on the x-axis there are infinitely many rational points.

My Proof

Lemma 1. Between any 2 rational numbers there exist infinite rational numbers.

Let $b\ge a$ multiply $a$ and $b$ by $10^n$ where $p$ is the smallest integer such that $(b\times10^p)-(a\times10^p)\gt10$. Let $m=\lceil a\times10^p\rceil$ and $n=\lfloor a\times10^p\rfloor$. From this we get $a\le \frac{m}{10^p}\lt\frac{n}{10^p}\le b$. As $m$ and $n$ are both rational be are done

Cruz Caine
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  • Is this supposed to be a proof of the lemma, or is the lemma something you alreday have at your disposal? (Also, you could zbe a bit less wasteful and multiply only with some $N$ such that $N(b-a)>2$) – Hagen von Eitzen Feb 13 '24 at 14:51
  • You still need to show that there are two distinct rationals between any two numbers. And if you know how to show this, then the conclusion doesn't require your lemma - just recursive application of the above fact. Even one rational is enough. – freakish Feb 13 '24 at 14:53
  • @HagenvonEitzen , would this be better?

    Let $b\ge a$ multiply $a$ and $b$ by $p$ where $p$ is the smallest integer such that $p(b-a)\gt2$. Let $m=\lceil a\times p\rceil$ and $n=\lfloor b\times p\rfloor$. From this we get $a\le m \lt n \le b$ and we are done

     – Cruz Caine Feb 13 '24 at 15:13

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