Talk:Between two Real Numbers exists Rational Number

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Hello, please see that this proof is incorrect by noting that $m, n$ in $N$ implies that for any given pair of real numbers $x, y$, there exists a *positive* rational between them. Of course this is silly.

Good call. Very silly mistake, as you point out.
BTW you are invited to sign your postings on talk pages by pressing the icon above the edit pane containing a squiggle (second from right) at the end of your posting. This puts two hyphens and four tildes at the point of the cursor. When the page is saved, the MediaWiki software expands this into a link to your user page. Hence it is apparent who wrote what comment and it makes conversations on talk pages easier to follow as you know who said what. --prime mover (talk) 07:21, 4 October 2012 (UTC)

Just one case

Seeing as this has been a featured proof, I thought I'd better ask before charging ahead...

I suggest making the following change in the proof. Change

Let $M := \left\{{x \in \N: \dfrac x n > a}\right\}$


Let $M := \left\{{x \in \Z: x > n a}\right\}$

and use Set of Integers Bounded Below has Smallest Element. (I edited the bound from $\dfrac x n > a$ to $x > n a$ to make it immediately obvious that $M$ is indeed bounded below.)

This removes the need to make a case distinction based on the sign of $a$. Is it OK if I make this change? KarlFrei (talk) 06:50, 5 October 2018 (EDT)

If you think you have a better proof, then add that second proof. I cannot immediately see that changing the above would eliminate the need for the case distinction. So add your second proof and we can see whether we can merge them. --prime mover (talk) 07:19, 5 October 2018 (EDT)
Done. KarlFrei (talk) 07:42, 5 October 2018 (EDT)
Need to figure out a way of documenting the intent of the original proof without reproducing it in full detail. But as it was (I believe, I don't have immediate access to the actual source work) a published proof it would be lax of us not to have it properly documented. --prime mover (talk) 08:28, 5 October 2018 (EDT)

I found the source:


As it turns out, there was an error in the published proof, as it assumed implicitly that x was at least 0. Editors here then found a way to fix this error / complete the proof. However, it seems to me that this new fix is much easier, so I would like to have this one instead.

In any event, we are definitely not under any obligation to record incomplete proofs. KarlFrei (talk) 08:50, 5 October 2018 (EDT)

We're under "no obligation" to do anything. But as you will pick up as you generally get to know the scope of $\mathsf{Pr} \infty \mathsf{fWiki}$, we like to acknowledge our sources to the extent of reporting in full and complete detail when they make a mistake. --prime mover (talk) 09:19, 5 October 2018 (EDT)
I'm getting the vibe here that you really, really, really hate to delete anything whatsoever that has previously been written.
If you want to explicitly report that the source made a mistake here, that is fine by me. But surely we do not need to keep this ugly fix (meaning the case distinction) that was put in? I mean, it works, but must we keep everything that our editors come up with? KarlFrei (talk) 09:31, 5 October 2018 (EDT)
Because that's what we do. --prime mover (talk) 10:52, 5 October 2018 (EDT)
... and I too have found the source, on my shelf of immediately-accessible texts:
Oh I see, you have a newer version... KarlFrei (talk) 03:53, 9 October 2018 (EDT)
... and from the "Preface to reprinted edition" in that same volume:
I am grateful to all who have pointed out errors in the first printing (even to those who mentioned that the proof of Corollary 1.1.7 purported to establish the existence of a positive rational number between any two real numbers). In particular etc.
Wow, he really did not care about this part of his book at all. KarlFrei (talk) 03:53, 9 October 2018 (EDT)
You see the damage to the paper through the words "otherwise $n x$ would" in the above scan? That's a result of when I tried to tear my copy of the book in two some while back. --prime mover (talk) 08:55, 9 October 2018 (EDT)
The proof can stay as is for now. --prime mover (talk) 12:57, 5 October 2018 (EDT)