Yesterday, I had a student come up with what I think is a really really good idea, pretty much out of nowhere. I’ll look to write about it but it makes sense to first have a couple warm-up posts. To begin, here is an inequality proposition to ponder as you begin the school holidays:
Suppose that 
and 
are real numbers, and suppose that for every 
, we have
![\[\color{RawSienna}\boldsymbol{a\leqslant b + \epsilon\,.}\]](https://mathematicalcrap.com/wp-content/ql-cache/quicklatex.com-4847ea67ad6739c7e34d09ff99f0835b_l3.svg "Rendered by QuickLaTeX.com")
Prove that 
.
University guys, please keep quiet for now. In particular, some of my former uni students may recognise this proposition, and may remember the name (which is the motivation for this post), although I think the main interest here may be different); if so, these students should wait to comment and let the school guys have a go.
UPDATE (22/06/26)
I’ve written a follow-up post, here.
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Hi Marty, just a quick thought, is this something we can use contrapositive?
You can use whatever you wish (except CAS). Do you qualify as a uni guy?
Maybe not? I was not maths background.
Yeah, but you’re Chinese not-maths background. I’m not sure that counts.
Do not read if you do not want your own thinking journey to be possibly spoiled.
Thinking Part 1:
Sees
. Instant wince.
Thinks about three different cases:
less than 
,
equal to 
and
greater than 
.
•
•
•
Two of those cases are clear and one is not.
Tries a specific case for
.
and 
.
should never be true.
that makes this work.
.
But I can find an
For any
Mmm, Marty must have got it wrong.
Thinking Part 2:
Surely Marty did not get it wrong.
Re – reads the question. Scratches head.
Phones a friend, discussion reveals a subtlety that was not noticed when reading the question: “suppose that for every
“. Scratches head. Scratches head. Scratches head.
So
for every 
, not an 
of my choosing. Mmm.
So focus on
.
Thinks about
.
Satisfied; decides to get ready for the footy.
Go Dees.
Thanks, storyteller. Obviously your conclusion to Part 1 was absurd, violating The Universal Law of Marty’s Correctness. And, it’s helpful to have some friend to phone.
While your support would have been nice, it was not needed! 76>62. 🙂
Rest of thinking:
If
If
With hindsight, clearly you should have chosen
and 
.
Thank, storyteller. That works.
You don’t need the “If
” bit, since this is what you’re trying to prove. So, it’s not a “cases” thing: just a proof by contradiction. (Anybody who says “proof by contrapositive” will get shot.) Your contradiction bit is fine, although stylistically a little wonky: it’s preferable, as others have done, to note a specific 
that stuffs things up, rather than your more vague “there is some 
“.
Had I the foresight to have chosen
and 
then I could have also tipped Carlton!
Glad it works.
FWIW, I was not trying to share a formal proof, but rather a summary of the exact thinking I did.
When I read the question, my mind auto-piloted to Proof by Contradiction Island. It is interesting to wonder why. I did not stay there long, as I did not see an immediately obvious way to proceed. Had I, or had I looked up a proof, I imagine I would have been very satisfied, even proud and myself, posted it and moved on to the footy. But that was not to be.
Then I found myself trying to properly understand the situation described by the very formal wording. That led me to plug in a few numbers; that lead me to think about cases.
As it turns out, cases seems like the most natural way to think about it, at least based on my thinking trajectory.
After reading your response, I went on another thinking journey, specifically about proof by contradiction.
For background, I have taught, at both school level and competition camp level, courses on inequalities.
The thinking I shared explains why
does not work and also shows why 
does work, which is what I was trying to achieve with my thinking when I did it.
If I was to prove the result using proof by contradiction, then I would be accepting that if I could show
failed then without further thought 
is correct. Perhaps for the first time ever, I really thought about this, and what followed was interesting-to-me.
Had I used proof by contradiction, it felt like I was just accepting , from the question, that
was correct. But how did I know that to be so? If felt like I did not.
So I did some more foundational thinking about proof by contradiction.
Consider proving that
is irrational using proof by contradiction.
can not possibly be rational then it must be irrational.
The proof is founded on the fact that we are comfortable that all numbers are either rational or irrational and if we can show that
Now say the same thing for the context you posed.
or 
and that if we can show that 
can never result in 
, then 
will always result in 
.
The proof is founded on the fact that we are comfortable that when all pairs of real numbers a and b are compared, either
These two situations feel very different–to–me.
The thinking I did left me convinced that
will be true for all 
and nothing else.
Contrast that with the case where I was perhaps taking a course where I was learning how to “do” proofs about “whatever”. And just did the proof, and did not do the thinking I did.
I am not an expert in any of this, just a hacker, but it feels like maybe I am currently aligning more with the constructive philosophy of mathematics rather than the classical one. 😂
I am sure this is now in the category of TLDR, but I will keep going.
I also pondered all of this in the context of I have to fill up 1 hour with an extension class next week. Would I use it?
As it happens, I do not have to do this, but I am almost motivated to wrestle a class from another teacher to give it a try.
Why am I almost motivated?
Well, if they could do the thinking I did, then that would be a very good thing.
It is unlikely, however, that they would do the thinking I did with the stimulus I had.
Let’s assume the students will be in Year 8.
As presented here, it would seem impossible to believe a typical Year 8 student in an extension class could easily decode what is written and so it would be hard for them to attribute any meaning/value/.. to it. So, it would be unlikely they would be able to/want to devote thinking time to it. To them I doubt it would even be an attractive puzzle.
So how might it be presented to provide them with the opportunity to do the thinking I did?
Marty chooses any two numbers. We will call these numbers
and 
.
he adds any positive number, 
.
to 
.
To
He then compares the size of
He repeats this process lots of times (Marty is a special guy), and claims that by studying the data generated, he noticed something interesting.
Be Marty.
If I try it, I will report on how it went.
Thank you very much, storyteller. It turns out that
, not 
, but most everything else rings true. (Your “Had I used proof by contradiction, …” paragraph didn’t make sense to me.)
I’m writing a separate update post, where maybe people will wish to continue the discussion. But, some reply here.
Your pondering in various modes points out there are quite separate questions to consider:
1) How does one think about this problem, to come up with a proof.
2) How does one write the proof?
3) How does a teacher/lecturer write the proof for the student reader?
4) How does a teacher/lecturer teach the thinking to get to the proof?
These are not the same, and it is critical to appreciate this, and none are particularly obvious.
Your thought of trying to get something like this across to a (smart) Year 8 class is hilarious-ambitious. I wan’t at all presenting this question to suggest it as suitable for school classroom discussion, although it would fit for a *testing* discussion for SM12 students doing proof by contradiction. Still, I agree with the sentiment, that one should always be stretching, and stretching the students, to get across deeper ideas/approaches when the market will bear it. And, yes, I think some sense of what is going on with this proposition can be gotten across to smart school students, in the same kind of way that the idea of suprema and infima (rather than just maxima and minima) can be gotten across.
A little tweak to the student stimulus is needed.
Marty chooses any two numbers. We will call these numbers
and 
.
he adds a range of positive numbers, 
.
to 
.
To
For each p, he then compares the size of
He then repeats this process lots of times (Marty is a special guy), and claims that by studying the data generated, he noticed something interesting.
Be Marty.
Yes, I get the sense of how you’d look to present it.
Wearing my teacher-hat (which itches and I want to throw it away, but cost of living etc…) I can imagine giving this out in the staffroom and receiving one of two responses:
1. Isn’t it just obvious? Why do you want to prove that?
2. Have you been talking to Marty again?
I think both are reasonable questions.
It’s not why I thought to post it, but I think it is reasonable to ask whether or why this is worth proving (as well as how).
I’m fairly confident on the “how”, it is the “whether” that I’m intrigued by but I think that for you to answer such a question would be akin to giving away the content of the next post prematurely.
Actually, no. I had a rather eccentric reason for posting it and then, only after posting it, did it occur to me that the problem was interesting from the “why bother?” perspective you raised.
The inequality to true for all ε > 0. I wonder if taking the right hand limit of each side of the inequality as ε → 0 would be along the right track.
Hi, Potii. In some general spirit, the proposition is consistent with what you are thinking, but what you wrote cannot be correct. One takes a limit of a sequence or a function, but there is nothing like that given to you here.
yep that makes sense
Contra, by the book
Or reductio ad abs
Yes, this was my reaction as well. Assume for reductio that a – b > 0, then set epsilon equal to (for instance) (a-b)/2 and contradiction follows.
(As for the cryptic comment in the post, is this proposition a consequence of the Archimedean property of the reals? If we could have both a ≤ b + epsilon and a > b, that would require a – b to be a positive infinitesimal).
Thanks, SRK. No, my interest in the proposition is much more pedestrian. Not meaning to be cryptic, just too tired to update the post. (Send beer.)
Suppose a>b. Then a= b+ c, for some c>0.then a (< or =) b+eps leads to
b+c (< or =) b+eps leading to
c (<or = )eps, which is false for, say, eps = c/2, so univ quant is false. c.b.d.u.
Thank you, Banacek. Correct, of course. What the hell is c.b.d.u. ?
Normally w.z.b.z. ,but this popped up via a friend. Beyond Latin. QED anyway.
And, um, what is w.z.b.z ? I gather it is “QED”, but how?
L.O.T.E.
Huh. Today I learned something.
Thanks to everyone for their thoughts. I’ll update the post soon(ish), with a tidier version of Banacek’s proof, and an explanation of why I posted the thing.
Although Red Five’s pondering of “why bother?” is not the reason I posted, I think it is well worth pondering, and I’m curious about what people think. Mathematicians will be familiar with and (usually) appreciative of this kind of thing, but what about you school guys? Is it at all interesting to prove such a thing, or does it just seem fussy and pointless?
I find this foundational stuff interesting, and I think now and then it can be worth bringing up with your students. But at the high-school level I think the focus is more on getting students comfortable with doing proofs of inequalities that don’t strike them as completely obvious / trivial. For instance, even getting students to give acceptable proofs of x^2 ≥ 0 can be a bit challenging, because they tend to regard this statement as trivial – so you often get circular arguments or “proof” by example.
I suppose the counter is that maybe we should introduce this sort of stuff in a gentle way, at least for those students who’ve got a strong chance of going on to study analysis at university, just to ease that transition.
I will answer your two questions from experience at
schools I have worked at over the last 
years:
1. Is it interesting? Yes, absolutely.
2. Do people care? Rarely and nowhere near as much as they should.
See comment above.