3
$\begingroup$

tldr: There is a simple intuitive definition of a limit for monotone sequences, and I suggest that it can be used to motivate the (more complicated) standard definition. I am asking for feedback on my suggestion. Do you think it can form a good motivation to present the students?

Edit: My target audience are students who are taking a rigorous calculus course, where full proofs and rigorous arguments are the center of the course. In particular, more than a third of the course is devoted to sequences and their limits. (Subsequence, Cauchy sequences, Bolzano–Weierstrass theorem etc). The course begins with sequences, and only after about 6 weeks we move to treat functions.

I suggest an approach for motivating the definition of a limit of a sequence.

What are good ways to help students understand the nested quantifiers in the definition of limit

While thinking of the definition of a limit of a sequence lately, I had the following thought: Naively, there is something which might appear unsatisfactory with the order of the quantifiers:

For every $\epsilon>0$ there exists $N$ such that...

Intuitively, we would like to say that "as $n$ increases, $a_n$ tends closer and closer to the limit $L$".

So very naively, we would like something in the spirit of $$n_1 >n_2 \Rightarrow |a_{n_1}-L| < |a_{n_2}-L|, \tag{1}$$

which suggests first choosing $n$, and then choosing $\epsilon$.

Of course, this specific formulation $(1)$ is bad, since it assumes monotonicity, and it also does not imply that $|a_n-L|$ tends to $0$, it can tend to $17$ instead. (it implies that $a_n$ tends closer and closer to $L$, but not necessarily arbitrarily close to it).


The point is that the standard definition can seem "intuitively backwards".

Question: Do you think that the following idea provides a good justification/motivation for the way we define the limit concept? (perhaps in honors courses).

  1. First, we define what does it mean for a monotone sequence to converge. This can be done in a more intuitive way, similarly to $(1)$, which produces something which is equivalent to the standard definition. (see definition below).

  2. Second, we use the Sandwich theorem as a guiding principle: It is intuitive to say, that whatever notion of convergence we want to have, it should obey the Sandwich theorem.

Now, one can prove that a sequence $a_n$ converges to $L$ (by the standard definition) if and only if there exist monotonically increasing $b_n$ and monotonically decreasing $c_n$ such that $b_n \le a_n \le c_n$, and $b_n,c_n \to L$.

For example, one can define $$c_{n} := \sup_{k \geq n}a_{k}, b_{n} := \inf_{k \geq n}a_{k}.$$

Then $$ \lim c_n =\limsup a_n = L =\liminf a_n =\lim b_n. $$

(One can prove this claim directly, over any ordered field actually, without assuming and using the existence of infimums or supremums, but never mind that for now).


To summarize, I suggest:

  1. Define convergence for monotone sequences (below).
  2. Show that a sequence converges via the standard definition if and only if it is bounded by monotone sequences which converge to the same limit (via definition $1$).

The point is that after we have done this, we keep only the standard definition, which this reasoning "confirmed", since it's much more concise and easy to work with, then constructing bounding monotone sequences every time we want to prove a limit etc...


My definition of convergence for monotone sequences:

We say a monotone sequence $a_n$ converges to $L$ if

  1. $\forall n_1,n_2 \in \mathbb{N} \,\,\, n_1 >n_2 \Rightarrow |a_{n_1}-L| < |a_{n_2}-L|$.

  2. $\forall \epsilon >0$, there exists $n \in \mathbb{N}$ such that $|a_{n}-L| < \epsilon. $

Item $2$ enforces the arbitrarily closed (without it there is no uniqueness).

$\endgroup$
4
  • $\begingroup$ To be clear, I do not (necessarily) advocate that we should present this reasoning to our students, since it's not trivial, and will take time that we usually don't have in calculus courses. However, I found this motivation pleasing for myself, and maybe it could be useful for Honors courses or something like that. $\endgroup$ Nov 24, 2022 at 16:18
  • 1
    $\begingroup$ @SueVanHattum Thanks. I clarified my intention. Basically the point of the question was to get feedback about the suggested approach for motivation. I am not sure if this type of question is accepted here though. $\endgroup$ Nov 24, 2022 at 17:15
  • $\begingroup$ Thanks for your comments. Actually, I was specifically interested in feedback on my suggested approach. I would be also interested to see some other approaches as well, but I think that this is a different question. (I edited the question and title). Anyway, I am interested in ways to motivate the definition, which is related to but a bit different than understanding the definition. $\endgroup$ Nov 24, 2022 at 17:37
  • $\begingroup$ Did you explain in this question what your objections are to the traditional limit definition quantifiers? I don’t see why we would want to go to a lot of trouble to "fix" something that isn’t broken. For all epsilon there exists $N$ is fine by me. $\endgroup$ Nov 26, 2022 at 0:57

6 Answers 6

9
$\begingroup$
  1. The concept of a limit has nothing to do with the order on $\mathbb{R}$. The standard definition of a limit of a sequence generalizes, almost verbatim, to sequences with values in $\mathbb{R}^n$, metric spaces, and topological spaces, which lack order structure.
  2. Perhaps all examples of convergence that the students naturally encounter are monotone. For example, the limits in the definition $\lim_{x\searrow x_0}\frac{f(x)-f(x_0)}{x-x_0}$ and $\lim_{x\nearrow x_0}\frac{f(x)-f(x_0)}{x-x_0}$ for the right and left derivatives are (eventually) monotone for any analytic $f$. In that sense, I think many students might only have a mental picture for monotone convergence. The problem here here is that while $a_n$ gets closer and closer to $L$ as $n$ grows is a wrong heuristics for limits in general, it is what happens in most examples.
  3. To motivate the more general setup, you may give an example of an infectious disease that's being eradicated: there might be occasional outbreaks, the overall incidence rate still tends to zero. Or decaying sine-wave sound signal. Or you may just say that the general setup simplifies the proofs, as we don't need to be bothered with checking monotonicity all the time.
  4. I feel that the pedagogically right way to treat this problem is to simply discuss honestly the difference between the two scenarios. We don't really care what happens with $a_n$ on a step-by-step level, it may occasionally backtrack, as long as it does not wander too far away. From there, the standard definition flows rather naturally.
$\endgroup$
3
  • $\begingroup$ Re (2): What about $\lim_{x \rightarrow 0} x \sin (\frac{1}{x})$? That's not monotone no matter how much you zoom in, and I'm pretty sure that's an analytic function... $\endgroup$
    – Kevin
    Nov 25, 2022 at 22:28
  • $\begingroup$ @Kevin, that function is not analytic at zero. $\endgroup$
    – Kostya_I
    Nov 25, 2022 at 22:30
  • 1
    $\begingroup$ Fair enough, but it's still a "natural" limit that you can reasonably show to Calculus I students. $\endgroup$
    – Kevin
    Nov 25, 2022 at 22:31
7
$\begingroup$

I don't like it. Had a hard time following it. Just tuned out. Yes, I'm not a Ph.D. in math. But neither will be the target students. You should have won me over. You didn't. I have the IQ to follow this stuff. But you didn't pull me along. This is not being flippant. You will have the same issues with students if you approach them as you approached this forum.

I think you are better off being truly motivational (intuitive, loose, colloquial, pictorial) when you want to be "motivational". And as an adjunct and on top of the rigorous approach. Not to supplant it. A supplement. You seem to want to present a slightly different (non conventional) approach to the rigourous explication and see this as "motivation". It's not.

Also, it feels very "product forward" or even "pet idea from R&D forward", rather than market back. I urge taking the opposite approach and looking at the market more. Where do actual students struggle, with what tendencies and at least a semiquantitative ("many" versus "few" students) observation. Not hypothetical students. Real ones in the wild.

Also, consider multiple ways of correcting the issues. Not just an alternate rigorous explication or even an intuitive one, but things like drill and explication and interaction and personality of the teacher. Remember teaching involves humans.

From an intuitive (conceptual, motivational) sense the idea of "getting closer and closer to" is a good starter concept of a limit. OF course this does not need to be monotonic. So after discussion of say a decaying exponential, you can draw some sort of collapsing sinusoid and show that some dithering (actually use this word, remember we are in "motivation" land) is good enough.

Heck, just draw the graph and get them to consider the view.

enter image description here

AND! Note you don't need to pre-emptively clarify this when first discussing limits. You can do this in a progressive manner. One view "closer and closer to" and then the corrective upgrade.

I think that the actual practice of rigorous proof in research papers (and competitive discussion with speakers in academia) makes neophyte academic teachers think that they need to avoid any progression of ideas. For fear of looking or being wrong This is not correct, is not how people (real human people with limited meat brains adapted for fighting and fucking, not rule based silicon assemblies) progress in training. Think about your limit...it does not have to approach a goal monotonically. Neither does your teaching!

$\endgroup$
3
$\begingroup$

I think the answer depends on the meta question: why are students learning the definition of a limit?

Here are some reasons I can think of:

  1. Because students are taking an intro-to-proofs class and limits give them something to "make proofs" about.
  2. Because students will take further classes that use $\varepsilon$-$\delta$ proofs and you want them to have familiarity with the proof format.
  3. Because students are taking an analysis/advanced calculus course where the theme of the course is "prove everything you already know".
  4. Because you want the students to "feel" the difference between the original non-proofs formulation of calculus and the modern formal framework.
  5. Because you want them to be able to answer "tricky" questions where reliance on their intuition would produce the wrong answer.
  6. Because your department mandates that "proofs about limits" show up on the exam.

Each of these goals might suggest a different way to teach limits. If you can state what your goal is, I think it would be easier to answer whether your proposal is "good".

$\endgroup$
2
  • $\begingroup$ Thanks, I agree with you. I edited the question. My target audience are students who are taking a rigorous calculus course, where full proofs and rigorous arguments are the center of the course. In particular, more than a third of the course is devoted to sequences and their limits. $\endgroup$ Nov 25, 2022 at 7:53
  • $\begingroup$ A simply reason would be: perhaps one shouldn't expect their students to blindly trust whatever theorem thrown at them without proof.... But yes, tailoring teaching to goal is good $\endgroup$
    – Argyll
    Nov 26, 2022 at 4:03
3
$\begingroup$

While monotone behavior is important in analysis (such as the monotone convergence theorem in measure theory), I think you should forget the emphasis on monotone behavior and just be honest with the students that the way we pronounce notation like $\lim_{n\to \infty} a_n = L$ or $\lim_{x\to a} f(x) = L$, namely “the limit as $n$ tends to $\infty$ of $a_n$ is $L$” and “the limit as $x$ tends to $a$ of $f(x)$ is $L$”, is backwards from the idea behind those concepts when they are made rigorous. It would have been better if we pronounced the notation as “the limit of $a_n$ is $L$ as $n$ tends to $\infty$“ and “the limit of $f(x)$ is $L$ as $x$ tends to $a$.” Offer a reason the precise definition leads us to quantify on the limit value (for all $\varepsilon > 0 \ldots$) first: it is in relation to how close we want to be able to get to the limit that we determine we are far enough along in the sequence variable $n$ or function variable $x$.

That is also why it gets even more subtle to prove convergence of a sequence or function when there is no proposed explicit limit value $L$, since the definition of a limit is not immediately applicable. That includes proving convergence of infinite series, where verifying convergence often precedes the estimation of the value of the series. I recall being confused when first learning convergence tests for series in the calculus, because here we were showing something converges without knowing ahead of time what it converges to.

The order of quantifiers is super important and subtle. Historically, it tripped up even prominent mathematicians. The difference between pointwise convergence and uniform convergence is the exchange of two quantifiers, and it is related to how mathematicians in the 19th century mistakenly thought they had proved an infinite series (convergent at each point) of continuous function is continuous.

$\endgroup$
3
$\begingroup$

Here is a simpler definition of limits which works better for monotone sequences, and has value in motivating further investigation of the concept of limits:

If a sequence $x_1, x_2, ...$ is monotonically increasing and bounded above, then $\lim x_n$ is the least upper bound of all $x_i$.

That is, the limit is a number $x$ such that $x$ is an upper bound (i.e. $x \ge x_i$ for all $i$) and $x \le x'$ for all upper bounds $x'$. This can be shown to be equivalent to the standard definition (using epsilons) for increasing bounded-above sequences.

I think this definition is a better one to present than something more ad-hoc, because the concept of a "least upper bound" (or "supremum") is an important one in Analysis. For example, my undergraduate Analysis lecturer used the existence of least upper bounds of non-empty bounded-above sets as an axiomatic definition of the set of real numbers.

Note that the concept of a "least upper bound" isn't a logically simpler definition — $x$ is a least upper bound if $\forall x' . (\forall i . x_i \le x') \implies x \le x'$, still with two quantifiers. But it's perhaps a simpler one to explain to students who aren't ready for formal statements with quantifiers, because it hides the quantifiers in the relatively-easy-to-grasp words "upper bound" and "least".

$\endgroup$
1
$\begingroup$

Two points to think about:

  1. It might be more intuitive to replace "from some $N$ onward" by "for all $n$ except finitely many".
  2. I often present the notion of a limit through a game: I claim that the limit is $L$. Challenge me: ask me to prove that all but finitely-many elements satisfy $|a_n - L|<0.1$, and then $|a_n - L|<0.001$. This is not convincing (show a counter-example), hence motivates showing for arbitrary $\epsilon$.
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.