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When writing $\delta$-$\varepsilon$ proofs, it's common that the ''natural'' choice of $\delta$ leads to the final inequality in the form, say, $|\ldots| < \varepsilon+\varepsilon+\varepsilon$ instead of $|\ldots| < \varepsilon$. It's always possible to correct the choice of $\delta$, but this has its disadvantages (see MSE links below). Instead, one can refer to the general principle (the ''$K$-$\varepsilon$ principle'') which says

      Instead of doing this, let’s once and for all agree that if you come out in the end with $2\varepsilon$ or $22\varepsilon$, that’s just as good as coming out with $\varepsilon$. If $\varepsilon$ is an arbitrary small number, then so is $2\varepsilon$. Therefore, if you can prove something is less than $2\varepsilon$, you have shown it can be made as small as desired.

However, why do standard textbooks never teach, or prove $\delta$-$\varepsilon$ using, this $K$-$\varepsilon$ principle?

Every analysis monograph in my library shirked this K-ε principle, including Abbott’s Understanding Analysis, Apostol’s Calculus, Bartle and Sherbert’s Introduction to Real Analysis, Michael Spivak’s Calculus, and Rudin’s Principles of Mathematical Analysis. Google found merely 2 books that tout this K-ε principle ―

  1. Arthur Mattuck’s Introduction to Analysis (1998), pages 39-40

  1. Frank Morgan’s Real Analysis (2005), pages 17-8. But Morgan doesn’t headline Mattuck's $K$-$\varepsilon$ principle as leading news, or teach it as a godsend to streamline $\delta$-$\varepsilon$ proofs.

Most students prefer this $K$-$\varepsilon$ principle for being MORE forthright and straightforward, because in actuality,

Math S.E. users kvetch that traditional $\delta$-$\varepsilon$ proofs are

user27289
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    It would help if the question started with what "$K-\varepsilon$ principle" is, since even Arthur Mattuck stated that it's not a common name. And I think the right question to ask here would be "At what point should the students be taught not to care about arriving at $\varepsilon$ exactly?". It's true that later in life people are satisfied with, say, $3\varepsilon$, or other expression that evidently tends to zero as $\varepsilon \to 0$. – Michał Miśkiewicz Oct 27 '23 at 12:45
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    Should we also have a $\sqrt{\epsilon}$ principle, and a $\sin(\epsilon)$ principle, etc? At some point you recognize that the expression you obtain as a bound can be made arbitrarily small, and beyond this it is just bookkeeping. – Steven Gubkin Oct 27 '23 at 15:09
  • @MichałMiśkiewicz You have my blessing to edit + improve my question. Thanks in advance! – user27289 Oct 27 '23 at 16:02
  • @Thierry Yes, you are correct. I am criticizing the insistence on ending a proof with <1ϵ instead of kϵ. You have my blessing to edit + improve my question. – user27289 Oct 27 '23 at 16:06
  • I learned that principle in high school. At university I think pretty much from the start we were (allowed to be) happy with a result like $3\epsilon$, maybe with an explanatory sentence why this is sufficient to complete the proof. – Nobody Oct 28 '23 at 08:11
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    @StevenGubkin I suppose the point is that being able to think in terms of more flexible bounds is very useful and analysis textbooks shouldn't demand your inequality chain finishes with $...<\epsilon$. But I'm not sure how else you'd teach it, because telling students to just get something that "looks good enough" will mislead them; comfortability with $\sin\epsilon$ bounds or whatever else takes a bit of time – FShrike Oct 28 '23 at 15:13
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    @FShrike I agree that being able to think in terms of more flexible bounds is useful, but I think you have to "pay your dues". We can easily adjust the argument to get the exact epsilon, so we are being honest with ourselves when we say we are "done" when we get $\sqrt{\epsilon}$. If the student cannot then they should work to get the exact $\epsilon$: it is a developmentally appropriate problem for them. – Steven Gubkin Oct 28 '23 at 15:20
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    @FShrike I have taught a few classes which were small enough that I could give students more individual attention. If I knew that a student was capable of completing the argument I would encourage them to write their proofs at a higher level. The students who could not I would ask for greater detail. – Steven Gubkin Oct 28 '23 at 15:24
  • Instead of saying "Let's all agree", one could explain why it's just as good. – Michael Hardy Oct 30 '23 at 02:40
  • @StevenGubkin Why yes, we absolutely should have a "$f(\varepsilon)$ where $f$ is monotonically increasing" principle. (You make it sound like you're dropping an absolute knock-down argument, but this kind of generalization is the bread and butter of mathematics!) – Daniel Wagner Oct 30 '23 at 15:25
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    @StevenGubkin If you have the time, I would very much appreciate it if you could expand your comments into an answer, in the form of a frame challenge. This is the answer I would give (and I might yet do so, if I can find the time...). – Xander Henderson Oct 30 '23 at 17:17
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    @DanielWagner: "yes, we absolutely should have a "$f(\varepsilon)$ where $f$ is monotonically increasing" principle." But that 'principle' is not true. The condition that one would need is that $f$ is continuous at $0$ and satisfies $f(0) = 0$. Yet, there is already a principle which encodes this type of argument (although it's not precisely the same): the composition of continuous functions is continuous. – Jochen Glueck Oct 31 '23 at 07:01
  • @JochenGlueck 'The condition that one would need is that f is continuous at 0 and satisfies f(0)=0'. This is not necessary in the colloquial sense, one only needs that $f$ is a surjective function from $(0,a)\to (0,b)$ for some positive $a,b$ see here. But one can prove that it is always possible to find a function such as the one you describe – Carlyle Nov 07 '23 at 11:09

8 Answers8

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My two cents on this is that it's related to two key issues that I stress in my undergraduate analysis course:

Key #1: Existence proofs have two parts: First you exhibit/construct the object, then you show that it works. When proving a quantified statement like $(\forall \epsilon >0 )(\exists \delta>0)(P(\epsilon, \delta))$, your proof should start by establishing an arbitrary positive real ("Let $\epsilon >0$") and then give an existence proof for $\delta$. So when writing a "proper" proof, you must give/define the $\delta$ ("Consider $\delta = 25\epsilon$") and then show that it works. You can't have $\delta$ being some sort of place holder.

Key #2: Satisfy the definition. If the definition of a limit is (yada yada yada)(stuff $< \epsilon$), then in a limit proof you'd better end with showing the stuff is less than $\epsilon$, not some other quantity. Again, two things to note: (1) a good instructor will point out that, yes, (stuff $< 4\epsilon$) is equivalent and (2) there are plenty of times when immediately after giving a formal definition you give a proposition that says something easier is equivalent. For this particular concept, it's my experience that formalizing the idea in the form of a $K$-$\epsilon$ Principle is overkill leading to student confusion. As the instructor, either accept a proof that ends with (stuff $< 4\epsilon$) because you have confidence in your students' understanding, or don't introduce a $K$-$\epsilon$ Principle that is only going to add an extra justification step in the proof that they are struggling with for other reasons.

Aeryk
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    +1. I also don't like this $K$-$\epsilon$ principle as a first method. The proof structure of an $\epsilon$, $\delta$ proof does not start with Given $\epsilon>0$, guess $\delta$ wildly. Instead, the proof structure starts with Given $\epsilon > 0$, find an appropriate value of $\delta$. There are perfectly valid strategies which start from $|f(x)-L|<\epsilon$ and lead, by a sequence of backwards implications, to $|x-L| < $SOMETHING$(\epsilon)$ and now you can say "We've found $\delta$". If SOMETHING is multiplication by some constant $K$, so be it. – Lee Mosher Oct 28 '23 at 15:59
  • "formalizing the idea in the form of a K-ϵ Principle is overkill leading to student confusion." Please expound why? Aren't students more confused with cluttered, unkempt δ's such as these? This K-ϵ Principle is user-friendlier, and forestalls cluttered δ's. – user27289 Oct 28 '23 at 18:32
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    @user27289 The $K$-$\epsilon$ principle would not actually alleviate the majority of the "cluttered, unkempt" examples you link to, because they have more going on than just a multiple of $\epsilon$ in the final answer. – Misha Lavrov Oct 30 '23 at 15:19
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    This $K-\varepsilon$ principle sounds like it would easily confuse students that are already not fully at ease with quantifiers, especially as it can lead them to use a $K$ which is it itself a variable that could depend on $\varepsilon$. – Stef Oct 31 '23 at 11:53
  • I understand the assertions that you are shouting out with bold faced text. I disagree with them. @user27289 – Lee Mosher Nov 07 '23 at 14:30
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I can't comment since I lack reputation, so I'll post an answer here. I largely agree with the other answers, and can say that every time I've taught Introductory analysis after a day or so a student will ask whether $2\epsilon$ is allowed and usually a very healthy discussion ensues and we talk about what the definition says and doesn't say. Because this is the first time students have likely proven anything like this, I ask them as first to be very specific and match the definition in their proofs. I find it's usually not that hard to get buy-in about this (after all, this is a class where students are asked to prove things that before they would have said were obvious: ("if $a>0$ then $1/a>0$", or "if a sequence of integers converges it converges to an integer"). I think students understand that the point of the class is to think about what needs to be shown, and how we can fit pieces together like a puzzle to do so.

One answer to your question might be "the way the words appear on the page is NOT the way the proof was written." The way I think many of us recommend students write these proofs is to leave gaps and to write in pencil expecting to change $N$ or $\delta$ in their proofs (if you want to see the style I mean, look here or here or here) and so in truth these are the last things the student will write and so writing $\delta=\epsilon/9$ isn't much more writing and has the benefit of matching the definition. I imagine many teachers use this method as well. A student who is reading a proof that "if $b_n$ converges to $3$ then $1/b_n$ converges to $1/3$" probably isn't surprised that this is true (it will feel obvious to them) but presumably they want to know why all the stuff is in there. And so long as the unkempt stuff is explained I think this can just be help for students trying to understand what is going on. Of course if no care is taken and everything is just splattered in the page, then alas. Part of my reason for making these videos is to show the process, and to show that one need not be afraid of these proofs.

As far as which is better - I think it might actually end up being more confusing to students to have different sorts of "rules" that are further removed from the definition. Sure it's ok for a proof to end $< 9\epsilon$ or even $N\epsilon$ if $N$ is "fixed". But what about $n\epsilon$ in a proof of convergence for a sequence $a_n$? or what if a student invokes the Archimedean principle to say "There exists an $N$ such that blah is $<N\epsilon$" at the end of their proof? These all look distinct to us, but surely seem similar to the student. Aren't they all just "A number times $\epsilon$?" And if your reply is that well the quantifiers are different, or that $N$ is bound to $\epsilon$ - then you are admitting that adding this extra rule has lots of thorny corners and isn't as simple as "a number times epsilon". I'm not saying that I don't talk about whether $2\epsilon$ is ok, but I let that be student-driven and almost always their reason is "oh it's easy let's just go change that thing and divide it by two." And then we get a chance to say "Can we do that? Why? Are you sure?" Reminding students of these sorts of things daily has benefits that I think outweigh the benefits from allowing constants. I guess I am saying that spending 20 seconds each day reminding students what freedoms we have (or don't have) is better than spending 5 minutes "once and for all" to prove some lemma taking care of these things.

I once needed a bank letter for a visa application to affirm that for each day in the past twelve months my balance was at least \$2000. The banker wrote me a letter saying "Adam's average balance over the last twelve months was \$5,000". I had to explain that this was not sufficient and that a bureaucrat could certainly deny my visa for this reason. I don't think he understood the distinction. Indeed he said "but \$5000 is bigger than \$2000." My point is that when we're first learning things, things that may seem "adjacent" to some are actually not adjacent in the ways that matter.

As a final note - in my school, a lot of CS majors take real analysis and I emphasize that writing proofs in real analysis is very similar to writing a computer program. If your code is supposed to return an integer, but returns the string "1" this is actually a problem. Perhaps now I realize that maybe bureaucrats are also like computer programs.

Adam Boocher
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    Yes exactly! Students will find it hard to distinguish between K as a constant vs K as a variable – whoisit Oct 29 '23 at 18:27
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    "Perhaps now I realize that maybe bureaucrats are also like computer programs." Bureaucrats & groups of bureaucrats are computers. It is bureaucrat job descriptions that are programs & programming is writing the job descriptions. That has been the best way I have known for decades of explaining to non-programmers what programs & programming are like & how easily one can make an error that causes the whole system to grind to a halt or spew out nonsense. – philipxy Oct 30 '23 at 10:37
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    Your banker has a better sense of wealth than your visa bureaucrat, but your visa bureaucrat has a stronger sense of "following a rule to the letter" than your banker. Having an average balance at least 5000 likely makes you a more attractive visitor than having a balance over 2000 each day, although one doesn't guarantee the other. – Stef Nov 03 '23 at 12:28
  • "I had to explain that this was not sufficient and that a bureaucrat could certainly deny my visa for this reason. I don't think he understood the distinction. Indeed he said "but $5000 is bigger than $2000."" I am befuddled. After your explanation, did your banker finally grok the difference? Or still not? "Perhaps now I realize that maybe bureaucrats are also like computer programs." Pls elaborate this simile. Why "bureaucrats are also like computer program"? – user27289 Nov 04 '23 at 19:56
  • The banker did not understand - this is a subtle concept that the mean of 10 numbers can be over x, even if some (or most!) of the numbers are less than x. Why are you befuddled? Math is challenging and everyone is at a different place with it. The last sentence of my post was a joke. – Adam Boocher Nov 04 '23 at 21:10
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The goal of teaching $\delta$-$\epsilon$ proofs is not necessarily to make students as efficient as possible in writing $\delta$-$\epsilon$ proofs. The goal is to instill understanding, of which the ability to write $\delta$-$\epsilon$ proofs is a symptom.

If we add tricks that make it easier to write a proof "in order", starting from the first line and ending at the last line, then we are not doing our students a service, because proofs are not generally written in order. The typical thing to do, when writing a proof, is to explore the problem, see how the parameters interact, figure out what's going on (maybe by solving some equations on scratch paper), and then write a proof.

It so happens that $\delta$-$\epsilon$ proofs are a good place to teach this skill, because the exploration that needs to be done is relatively routine. It is routine enough that something like a "$K$-$\epsilon$ principle" can be invented to eliminate the exploration. But we shouldn't give in to this temptation, because then the students will be stuck learning how to explore later, on a harder problem.

In addition, a real analysis class is often the students' first class in formal proof-writing. They are learning skills like "if the definition says 'there exists a $\delta$', then I will have to figure out what $\delta$ to choose to satisfy the definition, and pick that at the beginning of the proof". We should not muddy the waters while the students are still learning this basic logical structure. Once they are comfortable with it, then (if they are still writing $\delta$-$\epsilon$ proofs) we can teach them how to get away with being sloppy.

Misha Lavrov
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I think there are very few textbooks that concentrate on how to write proofs. The $K$-$\epsilon$ principle is a bit helpful in reading proofs, but it usually isn't too hard for an experienced author to choose a $\delta$ at the start in order to get $\epsilon$.

Where the $K$-$\epsilon$ principle is invaluable is for a student who is learning to write $(\delta, \epsilon)$ proofs and can't see which $\delta$ to choose at the start. (Especially if they are writing these proofs on timed tests!)

I agree with the OP's implication that it would be a good idea for more books to emphasize this, but this is my literal answer as to why I think they don't.

Also, "$K$-$\epsilon$ principle" isn't a very mnemonic name, and I've never heard it before. Perhaps the lack of a catchy name is part of the problem?

David E Speyer
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  • "I think there are very few textbooks that concentrate on how to write proofs." Truly? See https://math.stackexchange.com/a/4712160, https://math.stackexchange.com/a/4741235. "Perhaps the lack of a catchy name is part of the problem?" Recommend a few catchy names please? – user27289 Oct 28 '23 at 01:03
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    In the German system, the first semester real analysis class usually also doubles as the "introduction to proofs" class. But in that context I would be even more disinclined to teach such a "principle". For the "be able to write a mathematically correct proof"-part, the nonlinear thing of doing the calculation and then going back to choose the right $\delta$ is one of the key skills. And once you understand that, then figuring out this trick and why it works is a key part of understanding continuity. Putting it into a rigid principle that will just be memorized defeats both those purposes. – mlk Oct 30 '23 at 10:33
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By now, as somewhat experienced in what little of mathematics we use in economics and statistics (and limits are certainly a part), I recognize what the OP is raging about, and in practice, I always skip over it by the verbal "since we can make it arbitrarily small...". And I am fine with that, and everybody is fine with that.

But I am a user of mathematics at this point, not a student. And I suspect that the concept

"If $C$ is fixed and $\epsilon$ can be made arbitrarily small, then it doesn't matter if we arrive at $C\epsilon$ instead of $\epsilon$"

is not an easy one to grasp for students, because, mentally it "equates" (qualitatively) $C\epsilon$ with $\epsilon$, and this hits at the heart of what is truly difficult to understand, the consequences of the very small (and of the very large, analogously).

So I am not sure that this proposed principle would be helpful upfront as a tool to have. Realizing that this is so eventually, I believe would be a great mental satisfaction and intuitive milestone for the students.

Alecos Papadopoulos
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  • "as somewhat experienced in what little of mathematics we use in economics and statistics" Huh??? Economics and statistics use lots of math! – user27289 Nov 04 '23 at 19:52
  • @user27289 Certainly. Sometimes I may be writing more mathematical symbols than words in my economic research. The remark was not about volume but about coverage -whether, if ones overviews the various fields, subfields etc of mathematics, economics and statistics use many of them or not, in a qualitative sense. But, no contest, I will leave this to be assessed by mathematicians. – Alecos Papadopoulos Nov 05 '23 at 12:17
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Because inventing rules when teaching this topic does more harm than good.

The reason to make some principle a formal rule is to facilitate its operational use without remembering/understanding why is it true.

For example, to teach differentiation, one often just need to give the student a set of rules and a lot of practice with them. This is independent from them understanding the proofs of those rules; having to keep in mind the reasons behind Leibnitz rule when differentiating $x\sin x$ would be harmful.

This is, however, exactly the opposite of what we want to do when teaching $\epsilon-\delta$ proofs: it's not like the students have an engineering course ahead where they will need operational fluency with writing $\epsilon-\delta$ proofs. There's no other reason to teach those proofs than to make students understand them.

Kostya_I
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One might consider the purpose or goal of a particular book. Each author might have their own. [I am slow to answer because my real job takes time. Answer have been trickling in since I started to formulate mine. Forgive me if answers of some other's have escaped my notice or memory.]

To understand $\epsilon$-$\delta$ (real functions) or $\epsilon$-$N$ (sequences) definitions —

Then some attention to the technical difficulties in satisfying a definition probably seems appropriate to the author. Compare with Misha Lavrov's and Kostya_I's answers. A criticism I have of this approach is that sometimes students come out with a heavy, pedantic approach to analysis.

To present efficiently the theory of real analysis —

This seems Rudin's purpose, for instance, AFAIR. It also seemed the goal in Marsden and Tromba's Vector Calculus, at least as far their presentation of proofs. They tend to avoid $\epsilon$-$\delta$ when a hypothesis of continuity makes it unnecessary or by a form of reasoning I likened to an unstated squeeze theorem — not the dinosaur in typical calculus texts, but this:

If $|f(x)-L| \le g(x)$ and $g(x) \rightarrow 0$, then $f(x) \rightarrow L$.

The $\epsilon$-$\delta$ proof of this simple and at the same time intuitive (imho). This is essentially Steve Gubkin's $g(\epsilon)=\sqrt{\epsilon}$ and $g(\epsilon)=\sin \epsilon$ principles all rolled into one.

To teach students to prove theorems in real analysis —

This can be distinct from understanding $\epsilon$-$\delta$ proofs, as well as in addition to it. It also seems a goal of Rudin. This goal can be accompanied by an assumption that students have some general understanding how to construct a proof based on axioms, hypotheses, and other theorems. The author need not include a narrower focus on proficiency in applying the definition of a limit.

To give a gentle introduction to analysis —

This might mean the author wishes to teach students how to prove theorems in analysis but wants to avoid some of the technical difficulties in $\epsilon$-$\delta$ proofs, which the author assumes the student will master at some later date, should they ever need such mastery. Possibly that describes Marsden & Tromba, but I don't think they are trying to teach students to prove theorems.

The $K$ principle in the OP brings to mind the following version of the squeeze theorem (a "pointwise Lipschitz" condition):

If $|f(x)-L| \le M\,|x-c|$, then $f(x) \rightarrow L$ as $x\rightarrow c$.

Of course not all functions satisfy such a condition, but most "calculus-level" formulas do. Notable exceptions are roots like $\sqrt{x}$ at $c=0$, which are easy to handle with $\epsilon$-$\delta$. It probably doesn't belong in a general analysis course, but it's helpful in explanations at a lower level. And it prepares students for the inequalities found in the $\epsilon$-$\delta$ definition. It's not a point closely related to the OP's question, but sharing it felt more generous than not.

user1815
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This answer might involve more math than talk about math education, but nevertheless:

I am currently a student, and this precise issue has oftentimes plagued me, it started with me being frustrated in an exam when I would guess some expression for delta, and end up with something that is not quite epsilon, and then have to correct the whole ordeal as you describe, but recently I pondered it more deeply and realised that it's not just for expressions of the form $K\varepsilon$ that I can go back and apply a suitable correction, but also for an expression such as $\varepsilon^2$ although the correction is oftentimes more complicated, but then I asked myself, "How did I know that I can correct this, before seeing precisely how to do it?" Which lead me to realise that actually, if you end up with any surjective function (from $\mathbb{R^+}$ to $\mathbb{R}^+$) of $\varepsilon$, then in principle, you should be able to correct it, the "correction" that works, in principle, amounts to taking the specific $\delta$ which maps to the desired $\varepsilon$, guaranteed to exist by surjectivity. So actually, we can rewrite the condition

\begin{align} \forall_{\varepsilon>0}\exists_{\delta>0}\forall_{x\in \mathbb{R}}[|x-c|<\delta \implies |f(x)-f(c)|<\varepsilon ] \end{align}

As: There exists a surjective function $g:\mathbb{R}^+ \to \mathbb{R}^+$ such that for any $\delta>0$ and any $x$ we have

$|x-c|<\delta \implies |f(x)-f(c)|<g(\delta) $

Which generalises the "$K-\varepsilon$" principle.

To me this condition is much simpler, as you exhibit something first (the function $g$) before making a universal statement involving it, as opposed to making a universal statement, followed by an existential statement, but when I showed this to my lecturer, it took significant convincing, and eventually a very formal and overly detailed proof of equivalence of the two definitions to convince them that this is correct, although it is more abstract than the $K-\varepsilon$ principle, I think it illustrates an important point, which is that many educators probably are not too comfortable with things that don't look like what they learnt themselves. I also showed it to some of my friends, and they were mostly confused, which I think also illustrates that sometimes a more "elegant" definition is not the easiest to understand, in fact sometimes something which is "elegant" is just very "clever", and oftentimes when students start out with course material, cleverness to them seems like daunting magic, especially if they're not able to fully justify it without assistance. Anyhow, I know this doesn't quite answer the question, but I thought it was interesting

Carlyle
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