The Thrilling Epsilon Lemma

June 21, 2026

10:29 pm

10 Comments on The Thrilling Epsilon Lemma

analysis, proof, puzzles, Specialist Mathematics

Yesterday, I posted an inequality proposition for readers to ponder:

Suppose that $\color{RawSienna}\boldsymbol{a}$ and $\color{RawSienna}\boldsymbol{b}$ are real numbers, and suppose that for every $\color{RawSienna}\boldsymbol{\epsilon>0}$ , we have

$\color{RawSienna}\boldsymbol{a\leqslant b + \epsilon\,.}$

Prove that $\color{RawSienna}\boldsymbol{a\leqslant b}$ .

The discussion was very interesting, and thank you very much to the industrious and thoughtful readers-commenters. I had planned an update, but there turned out to be a lot that I thought worth writing, and so I decided on this separate post.

1. THE PROOF

We’ll come to the title of this post but, first, how do you prove the thing? Commenters on the previous post got it right, but let’s do it as neatly and as clearly (but not nearly as quickly) as possible.

It is not at all obvious unless you’re used to this stuff (see parts 3 and 4), but it turns out that you need to do a proof by contradiction.¹ So, suppose the conclusion is false, and that $\boldsymbol{a>b}$ .

Now, it is not necessary, and cannot be part of the formal proof, but a simple sketch (in the sane, non-VCAA sense of the word) greatly helps the intuition. So, if $\boldsymbol{a>b}$ then there is a gap between $\boldsymbol{a}$ and $\boldsymbol{b}$ , and let’s call that gap $\boldsymbol{d}$ , as pictured.How does this help? Well, the information we’re given is that

$\boldsymbol{a\leqslant b + \epsilon}$ for all $\boldsymbol{\epsilon > 0}$ .

But this cannot be true for an $\boldsymbol{\epsilon}$ that is smaller than the pictured gap. So, if we choose any such $\boldsymbol{\epsilon}$ smaller than $\boldsymbol{d}$ , that will give us a contradiction.

Specifically, let’s choose

$\boldsymbol{\epsilon = \frac{d}2\,.}$

Then, by our hypothesis, and noting that the gap $\boldsymbol{d= a -b}$ ,

$\boldsymbol{a\leqslant b + \frac{d}2 = b + \frac{a-b}2 = \frac{a}2 +\frac{b}2\,.}$

Subtracting $\boldsymbol{\tfrac{a}2}$ from both sides and doubling, we find that

$\boldsymbol{a\leqslant b\,.}$

That is, the assumption that $\boldsymbol{a>b}$ leads to the conclusion $\boldsymbol{a\leqslant b}$ anyway. And so (either way you cut, if you will), $\boldsymbol{a\leqslant b}$ , which is what we set out to prove.

1) A reminder, anybody who says “proof by contrapositive” gets shot.

2. THE THRILLING ε-LEMMA

As the title of the post indicates, this proposition is called the Thrilling $\boldsymbol{\epsilon}$ -Lemma. According to Google’s AI mode,

The “Thrilling Epsilon Lemma” is an affectionate, tongue-in-cheek term coined by Australian mathematician and educator Burkard Polster (popularly known for his YouTube channel Mathologer) to describe a foundational technique used in real analysis and measure theory.

I feel like Rodney Dangerfield: I don’t get no respect.

It’s not all Google’s fault. The lemma appears in lecture notes that I wrote,² and that live on QEDcat.com, Burkard’s and my website. And, Burkard is, of course, the (rightly) famous one. The lecture notes were for an honours subject, measure theory,³ which I taught for AMSI for many years.⁴

Why the name? The name is ironic, of course. The term “lemma” refers to a mathematical proposition that is not of particular of interest in itself but rather is used to prove a more central theorem(s). I decided to label the lemma “thrilling” because, although the lemma is decidedly not thrilling, it gets used roughly a thousand times in the measure theory subject. The lemma needed a distinctive name and “The Thrilling $\boldsymbol{\epsilon}$ -Lemma” was funny enough to stick, and was thereby really useful.

2) See Handout 2, p9, and Handout A, p3.

3) It is totally off the point but, while we’re here, if readers want to see the application of measure theory to group sex, they can go to 68:00 of Lecture 5.

4) AMSI eventually declared me persona non grata.

3. TEACHING THIS STUFF

My main purpose in posting on this was simply to point out the value of the title, of having a stupid-funny name for a pedestrian but really important lemma. So, I regard “thrilling $\boldsymbol{\epsilon}$ -lemma” as in the same category as BISS. It is a way to cut through the perception of sameness, to indicate that there is something important here that requires more attention than appearances might suggest. But, having had to explain the context, it seems worthwhile discussing a little how to teach and to write this stuff.

When I was a regular mathematician guy, my specialist teaching thing was in real analysis: broadly, the theoretical underpinnings of limits and calculus, and its generalisation to more abstract worlds. Real analysis is properly introduced as a very standard second year subject,⁵ and, if done properly, it is not easy. Students have to learn to prove statements, and very little of the proof in real analysis can be done in a recipe manner; the $\boldsymbol{\epsilon\!-\!\delta}$ stuff is notorious for a reason.There is properly very little distinction between the theorems in the subject and the exercises that follow; the “why” really is the core of the subject. In the measure theory subject, for example, the Thrilling $\boldsymbol{\epsilon}$ -Lemma is an early and very important exercise; it is an ingredient of so many proofs to come, students really need to understand what is going on, and nothing makes them understand it better than having them prove it (with help as required, of course).

One very difficult aspect of learning analysis proofs is to get used to the idea that the process of figuring out a proof is often very different from how the final, written proof should look. With more computational mathematics, and even with a lot of proofs, one begins with what is given and works it around until the desired goal is reached. With analysis proofs, however, it is common to begin with the conclusion and ask, “how can I get there?” Then, working backwards, one can see how to begin. But, when writing the formal proof, this discovery process is typically hidden; all that matters is the final, correct path, not the manner in which the path was hacked out of the jungle.

This also makes writing and lecturing for analysis classes a very tricky business. There is always a tension between trying to explain how to think about the analysis and how to write it cleanly once the thought is done. There are lots of judgment calls. I have my own style, which leans heavily into being chatty and wordy, so that the proofy proof bits are less daunting, but this verbosity has drawbacks. There is a good argument for just shutting up and giving the proofs.

Lots could be said here but I’ll just make one final note: it is inconceivable that one could use PowerPoint to teach real analysis in any proper manner. The subject matter and the manner of mastering it is simply too non-linear for a fundamentally linear medium such as PowerPoint. Of course that linearity argument does not automatically extend to the teaching of school mathematics, but it should also not be so quickly dismissed.⁷

5) A decent first year subject will have a hint of analysis, and a properly advanced first year subject can have much more than a hint.

6) Of course, mathematics departments can easily cheat, and simply fake the teaching of proof. So, the lecturer proves the main theorems while the students sleep, and then the students are awakened to work on some cookbook twaddle. The students learn nothing and are typically happy, oblivious to the fact that they have been cheated. The University of Melbourne did this for a while, although I believe it may have ceased.

7. I will have more to write on this soon (in the Marty sense of “soon”).

4. WHO CARES?

Should any of this matter for mathematics teachers, beyond their wanting to be a little more mathematically cultured? Not a whole lot, I think, and despite storyteller’s enthusiasm, I wasn’t suggesting the Thrilling $\boldsymbol{\epsilon}$ -Lemma as something to try on school students. But there is something there.

I do not think it is all that important that senior teachers know much of anything about the foundations of limits and calculus, and of mathematics generally, although of course it cannot hurt.⁸ I do think, however, that it is important that teachers are aware that such foundations exist, that this stuff properly nailed down is not remotely how it is presented in school textbooks. It would at least give the teachers the possibility of presenting school topics in a more nuanced, and more truthful, manner. It would also mean that the teachers would be more open to the reality of the textbooks and VCAA getting it wrong (and way too often). If nothing else, that would make my job a hell of a lot easier. It would be really nice if more teachers understood mathematicians like me, that we’re really not other-worldly pedants – at least not all of us – and that we have a deeper perspective on mathematics that is simply not being given due respect.

More broadly, an appreciation of what mathematics is, beyond how schools are instructed to present mathematics, would open teachers eyes to how schools might present mathematics. That is not at all to suggest that school mathematics should be theorem-proof; epsilons, for example, have absolutely no place in schools. But one can still be more thoughtful, to provide a little more meaning and more accuracy with the fundamentally intuitive and/or cookbook approach suitable in the school setting. In sum, teachers needn’t be masters of mathematical foundations, nor even have had much exposure to this stuff. But, they should be aware of the existence of this deeper world underlying school mathematics.

And, of course, the more prominent or more senior the person, the more important this is. Anyone pronouncing on mathematics within or for VCAA needs to know what mathematics is; anyone pronounced in the media to be a “maths expert” should be an expert in mathematics. But no one much cares, except for the mathematicians, who look on in horror at what is considered, and self-considered, to be an expert.

8. I’ve talked seriously with a very smart teacherish person about us co-writing and co-presenting a mini-subject, Analysis for Teachers. It would be a good thing, I think, and we haven’t given up on the idea. Just too much to do.

5. A FINAL STORY AND A BIT

If mathematicians and, much more importantly, mathematicians’ way of seeing mathematics, deserves more respect in school, it is also true that mathematicians must be respectful of those who know less mathematics. It is important, of course, as it is for all teachers, that mathematicians be empathetic towards their students, to appreciate the good work while guiding the students towards better work. This empathy is particularly important in a proof-type subject, where writing proofs that are both correct and well-written is extremely difficult; first attempts are almost invariably going to be a mess. But mathematicians within the shared space of mathematics education also have to be respectful in that space and, to say the least, that has not always been the case. Some mathematicians are assholes.⁹ This is a story and a half about a couple asshole mathematicians, and a non-asshole mathematician.

The first is a short story of a very smart mathematics teacher, who was on an AMSI committee sometime back. The committee was producing school mathematics of some type and the teacher gave his opinion that the draft materials were too formal and dry for a school setting. A mathematician on the committee then responded something like “Well, as a teacher, you may not appreciate the importance of rigour in mathematics.” This was simply bashing down, and stupidly: the teacher wasn’t objecting to rigour, but to the appropriateness of dry rigour in the school setting. And, although the mathematician was oblivious to the possibility, the teacher knew more mathematics than the mathematician and had about twice her IQ.¹⁰ To his credit, the teacher laughed the whole thing off: he didn’t hold the mathematician in high enough regard to be bothered.

The second story involves another smart mathematics teacher and is more directly connected with the epsilon business, and is hilarious. This teacher was involved with one of Melbourne’s “advanced placement” subjects, where students do first year university mathematics as a VCE subject; there are various models, but typically the subject is taught by the teachers, and the university administers the subject, including the setting and marking of the final exam.

This particular subject had a little bit of epsilons and deltas, and the teacher was not so sure he knew how to write such proofs accurately and properly. So, he wrote up his proof of an assigned question and sent it to the administrator, a tutor-type mathematician at the university, for her opinion. The mathematician replied that the teacher’s proof would have received zero marks. Now the teacher was not sure of what he was doing, but he was also not an idiot and he was pretty sure that his attempt was worth more than a zero. He decided to get a second opinion.

God knows why, but the teacher decided to email a mathematician at Princeton University. I think the choice was random but, in any case, he got onto a Fields medallist, asking for this guy’s opinion on his first year epsilon-delta argument, which the local tutor-type had awarded a zero. The Fields medallist duly replied; he suggested that the proof was not how a mathematician would write it, but it was basically sound. “Was it worth only a zero?” “Of course not.”

There you have it. And the moral of the story? I dunno. Expertise matters? Some people are assholes? Whatever you want.

9. I do not consider myself to be an asshole, but of course that is a minority opinion. Feel free to include me.

10. I know both the teacher and the mathematician well enough. The mathematician was also one of the people who shafted me at AMSI.

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10 Replies to “The Thrilling Epsilon Lemma”

Alex says:

June 22, 2026 at 1:24 am

My proof by contradiction wasn’t anywhere near as clean, but it was the same basic idea. But I then spent a long time trying to prove that a wasn’t necessarily strictly less than b (ie. that the result needed the ‘or equal to’), before realising i didn’t need to.

But if we’re in a mood of being intellectually generous to poor clueless teacher types, why *isn’t it* strictly a<b?

Reply
1. marty says:
  
  June 22, 2026 at 2:40 am
  
  Thanks, Alex. Writing clean proofs is *hard*.
  
  The short answer to your question is that $\boldsymbol{a}$ *may* be less than $\boldsymbol{b}$ , but it doesn’t have to be. So you may have, for example, $\boldsymbol{a=3}$ and $\boldsymbol{b=5}$ . But you may also have $\boldsymbol{a=5}$ and $\boldsymbol{b=3}$ : that is also consistent with the hypothesis. But this short answer doesn’t really give the intuition of why you cannot conclude $\boldsymbol{a <b}$ .
  
  For the intuition, think of one of those squeezing rooms in a thriller movie, where the walls are coming together. Every time you choose an epsilon, you’re squeezing $\boldsymbol{a}$ closer to $\boldsymbol{b}$ . But even at the “end”, using really tiny epsilons, you’ve only squeezed $\boldsymbol{a}$ and $\boldsymbol{b}$ together. Nothing is squeezing $\boldsymbol{a}$ to actually be on the other side of $\boldsymbol{b}$ .
  
  Reply
  1. Alex says:
    
    June 22, 2026 at 10:33 pm
    
    I think I have a similar intuition, only it only ever gets a and b close, but never lets them be equal.
    
    My backstop is that I can prove a≤b, but I can’t prove a<b and that's all there is to it, but I can't get my brain to feel it.
    
    Reply
    1. marty says:
      
      June 22, 2026 at 10:37 pm
      
      Any specific epsilon can only get you close. But all the epsilons together show that there is no “limit” (pun intended) to how close you can get.
      
      Keep thinking of the room with one wall sliding in. It will squish everything into the opposite wall, but it won’t make anything pass through that opposite wall.
      
      Reply
tom says:

June 22, 2026 at 2:55 pm

I feel Marty’s stories have given me permission to tell my own personal journey, all related to the epsilon/delta business of course.

1963, I was going into the standard science degree at Melbourne University. Somehow I learned that previously there was a wholly separate subject, the honours stream began in first year, but this had been considered to favour those lucky enough to get good teaching at school. So the new policy was to have a common first year. Nevertheless, those of us with first class honours in Year 12 were assigned to a separate lecture group.

Somehow my lecturer had the freedom to follow a traditional honours stream analysis course, a sequence of theorems, epsilon/delta limits, Dedekind cuts, the works. I discovered much later our some of the select class had transferred to the more congenial mainstream classes where there was little theorem/proof and much more problem solving. In my innocence I did not know that one was allowed to transfer.

In those days assessment was based on yearly exams. I walked into my final exam to discover that I could not do a single question. All questions except one were based on the mainstream syllabus. I staggered out early and wandered into the first pub in North Melbourne. Later in college I found others similarly traumatised. In the 60s we put this down to a basic principle: life’s unfair. There was no concept of appeals or suing.

I expected to fail and have another go. But the Department was clearly deeply embarrassed; I was passed with zero marks. This made things worse for me, as I decided that it was impossible to fail, so spent the next two years following my own interests, attending courses I was not enrolled in, and having a social life.

There was a general changing of the guard after that first year, with many from the select group transferred to the pass course for second year, and some of the other stream transferred to honours. Years later I met one of these, he said he had not felt himself good at mathematics and was surprised to do so well in first year. All this reinforced those staff in the Mathematics Department who had claimed that this showed the school results were not a good predictor of later achievement. This of course was face-saving for the department. Worse, they had just proved that they were better teachers than the schools.

Reply
1. marty says:
  
  June 22, 2026 at 3:26 pm
  
  Thanks, Tom. The University of Melbourne has a long and proud history of such arrogant stuff-ups. When I was there in the 90s, they were still arrogant, still stuffing it up and still oblivious to both. I think there is something in the sandstone that damages the brains.
  
  Reply
  1. tom says:
    
    June 22, 2026 at 5:32 pm
    
    Miner’s brain, not miner’s lung?
    
    Reply
    1. marty says:
      
      June 22, 2026 at 5:33 pm
      
      Ruling class miners.
      
      Reply
Barry Hughes says:

June 23, 2026 at 12:38 am

Once when I was teaching complex analysis to third-year students and a few postgraduate students, one of the latter asked me to teach the entire subject without use of proof by contradiction. I politely declined to do so.

It is true that some things that it may seem easiest to prove by contraction can be done more directly under duress, so here for the problem at hand is my attempt. To keep things totally honest I suppose I would need to have shown first without use of proof by contradiction that inequalities are preserved by adding the same number to both sides and by multiplying both sides by a positive number, but at my age life is too short to worry about that here.

If $a = b$ then it is also true immediately that $a\leq b$ and therefore if suffices to prove that if $a\neq b$ and $a\leq b + \epsilon$ for all $\epsilon>0$ then $a\leq b$ .

I choose $\epsilon = \vert a - b\vert/2$ and note that $a - b = s\vert a - b\vert$ , where $s = 1$ if $a>b$ and $s=-1$ if $a < b$ . Therefore I have

$a \leq b + \vert a - b\vert/2$

so that

$a - b = s\vert a - b\vert \leq \vert a - b\vert/2.$

Multiplying by $1/\vert a - b\vert$ (which is why I dispensed with the case $a=b$ earlier) I find that $s\leq 1/2$ . Therefore $s\neq 1$ , so $s = -1$ and $a<b$ , as required.

I remark that this approach applies equally well to prove that if $a< b < \epsilon$ for all $\epsilon > 0$ then $a\leq b$ . One would want to alert students that it is not safe to conclude here that $a< b$ if we have not explicitly ruled out case $a=b$ .

Reply
1. marty says:
  
  June 23, 2026 at 9:40 am
  
  Thank you very much, Barry. That is very nice, and an excellent illustration of why mathematicians (and some postgraduate students) get such a bad name.
  
  i think a good analogy to not using contradiction is, you can play tennis without a backhand but you’ll often find yourself trekking a *long* way out of the centre of the court to do so. On the other hand, it is easy for a young student who has learned and mastered proof by contradiction to go for it too quickly as a killer shot, when a simpler direct proof is on offer. It’s like a kid insisting on playing a two-handed backhand when the forehand is right there to be played.
  
  Reply