I'm teaching introductory real analysis at a large public university in the US. A common question from students is of the form

"Why can't I just do it like this?".

i.e. Often a student has come up with a not fully rigorous but quick or easy version of a correct analysis proof and then demand to know exactly "where" the mistake is.

A concrete example would be e.g. showing that the series $\sum_{n=1}^{\infty}(a_n + b_n)$ diverges to $+\infty$ say by using $$ \sum_{n=1}^{\infty}(a_n + b_n) = \sum_{n=1}^{\infty}a_n + \sum_{n=1}^{\infty}b_n $$ and then saying that $\sum_{n=1}^{\infty}a_n$ converges and $\sum_{n=1}^{\infty}b_n$ diverges to $+\infty$.

One of the difficulties of the teacher is that the student happens not to have made a fatal mistake... they have "got the right answer" and now you are telling them that something is still wrong. The issue of course this isn't a real equation; it's a nonsense equation like "$+\infty = 1+ \infty"$ and really they need to argue with the partial sums. i.e. The real mistake is usually just... "you've used this like its a general fact but it isn't a general fact"... and that usually seems like a weak answer to the student (in my experience). I think this is partly because until you are more experienced and have done analysis in an unknown setting, you are not used to deliberately considering all of the things that could go wrong, so they are focussed on the fact that in this example nothing went wrong so what's the big deal?

Clearly I can't be expected to come up with the exact explicit example that highlights where their thinking would have let them down, but sometimes it feels like I'm trapped into doing that (and coming up with an example live at the blackboard is tricky!).

Are there are any good tricks or tips to deal with this sort of issue when teaching analysis?

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    $\begingroup$ What is the problem with the concrete example? $\endgroup$ – Tommi Brander Dec 6 '18 at 19:33
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    $\begingroup$ This just isn't an equation. Both sides are "$+\infty$" and I think that when learning analysis, you should not be manipulating equations like "$\infty = 1 + \infty$". It like putting $\lim$ infront of everything you are manipulating and then in the end you are proved right... but to begin with its not actually a valid argument unless all the limits exist. $\endgroup$ – T_M Dec 6 '18 at 19:49
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    $\begingroup$ I wonder if asking "why" is the right answer. In some sense, what we're doing in analysis is saying "For this answer to be correct, you must show me that it satisfies the definition of ___. You have not shown me that it does, therefore its not correct." Asking why might lead them there: Why is $\sum b_n = \infty$? What does that mean? $\endgroup$ – Nate Bade Dec 6 '18 at 20:48
  • $\begingroup$ You could show how this additivity isn't true in general (use $a_n = (-1)^n$ and $b_n = (-1)^{n+1} = -(-1)^{n}),$ although a problem for you in this case is that (I think) it actually is true that if one series converges and the other doesn't, then the term-by-term series doesn't converge, at least, it's a fairly general rule in math that "nice" combined nicely with "bad" produces "bad". Since this is a real analysis class, they should be seeing and constructing proofs. Ask them to give you an $\epsilon$-$N$ proof of what they claim is obvious, and maybe they'll see it's not entirely obvious. $\endgroup$ – Dave L Renfro Dec 6 '18 at 22:57
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    $\begingroup$ @DaveLRenfro Yeah somehow this is the things that's unconvincing for the student. You are saying " 1. Your argument is a case of this general claim. 2. This general claim is false, here is a counterexample". But the student isn't thinking of a general claim, they are just manipulating whatever they see and then ending up with the correct answer. As a teacher we freak out because its clear to us that they subconsciously used some generally false claim, but they don't see it that way. $\endgroup$ – T_M Dec 6 '18 at 23:43

An "introductory real analysis course" might be first-year calculus or what is at my US university a course taken in the fourth-year. I assume the latter. In either course, the problem of $$ \sum_{n=1}^{\infty}(a_n + b_n) = \sum_{n=1}^{\infty}a_n + \sum_{n=1}^{\infty}b_n $$ might arise. Certainly, the fallacy of splitting a limit is discussed even in calculus, both when limits are introduced and in the case of the limit of the partial sums.

My reaction to the OP's question is, don't the students have to justify their steps? In which case, what's the justification for the above step? It's the writer's (i.e., the student's) responsibility to present a valid argument. This applies to first-year calculus as well, although I'm more lax about the student justifying some steps after the fact (which I realize is the very thing the OP wishes to address). But in a problem where a logically sound proof is asked for, I would be strict.

Let's say you ask for justification and the student responds because $\sum a_n$ converges and $\sum b_n$ goes to infinity. If that's a valid argument because there is some such theorem in the course, then say, "Good! The hypotheses for that theorem should be established before the theorem is applied." If there is no such theorem, then say, "Good! But you would have to prove such-n-such a lemma to complete the reasoning."

Let's say you ask, and the student replies, "Because it's true!" You could reply, "So is the proposition we're trying to prove. Why not just say because it's true at the start?" For many students this would be enough. For some, it leads to a much larger question of why are we proving all this stuff we already know is true. You probably don't want to have that conversation in class. Ideally, it would be addressed at the beginning of the course, or in an earlier course. At some point, the curriculum should bring students to the point of being beginning mathematicians, but mathematicians nonetheless. (For many, this happens before college, because they are the ones in calculus demanding to know why things are true.)

With respect to "I can't be expected to come up with the exact explicit example that highlights where their thinking would have let them down," I'm not sure I agree. I've certainly spent a good deal of time in preparing for courses thinking and searching for just such examples that highlight the need for each hypothesis. Textbooks have quite a few, so you don't have to do everything yourself. I have also been at an institution that emphasizes teaching, so I have had the time to do so. It has been very helpful, since teaching by example is so much more powerful than, say, proving by example.

  • $\begingroup$ Thanks this is helpful. It's somewhere in between. It's definitely not calculus, but typically the students are 3rd years or non-math graduate students. I would say that some are beginning mathematicians but many are not. I suppose some other context I left out of the question originally is that - yes -Im really thinking about this situation being "on the spot" e.g. in the middle of class or during office hours... Then they sometimes catch me off guard with a bespoke idea and I often do not have a prepared counterexample. $\endgroup$ – T_M Dec 9 '18 at 22:11
  • $\begingroup$ Indeed I think "why are we proving all this stuff we already know is true?" has perhaps not been adequately addressed (and this might just be my fault). But partly the course is sufficiently basic that we get very few concrete payback out of all of the care and effort we put in to building the theory. $\endgroup$ – T_M Dec 9 '18 at 22:13

The issue here is that the student is still trying to learn a fundamental property of mathematics as a field of study - that the truth of any claim can, in principle, be reduced back to a relatively simple set of axioms and relatively simple rules of inference. Furthermore, this is something one should be capable of doing to any claim one wants to make use of.

This is the rigorous stage of learning mathematics, according to the classification of Terence Tao: https://terrytao.wordpress.com/career-advice/theres-more-to-mathematics-than-rigour-and-proofs/

At this stage, the appropriate response to an uncertain claim is to provide a proof. If something is obviously true, then it should be proven from first principles or reduced to known facts proven from them. This is something you have to tell to the student, and also to demonstrate, until they understand the idea.

As for the specific example, the reasoning is correct on a moral level: A converging series is essentially just a constant (up to an epsilon of error when the index is big enough), which will not affect the convergence of the summed series, so the diverging part dominates.

The proof could go by reducing the problem to sequences and showing that the sum of a converging and a diverging sequence diverges (maybe only in the case the divergence is towards plus infinity). You want the sum to be eventually bigger than a fixed $M \in \mathbb{R}$, and the converging sequence is almost $A$ (up to some small epsilon that you can choose to be less than one) when the index is big enough, so consider indices so large that the diverging sequence is bigger than $M-A+1$.

Some arguments that might help the student see this:

  • What if you have a sum of two converging sequences, or two diverging ones, or one diverging to infinity and the other oscillating within some bounded interval, or one converging to plus infinity and the other to minus infinity, etc. Can they answer what happens in each of those cases?
  • There is a lot to remember unless they learn to prove things. It is a compression technique - large amounts of information are compressed to a much smaller number of proof techniques. This is especially true, since there will be more and more mathematics to master in further courses and studies.
  • Someone needs to know and verify which computational techniques and shortcuts are viable. You are training to become that one.
  • It increases one's confidence when doing mathematics when one can simply prove things whenever doubt arises. As a working mathematician I do this often for quite simple claims, even though I have a good idea what the correct answer is, because this helps to avoid simple mistakes.

One should remember that not everyone can learn this, or at least it is terribly difficult to learn for some people (non-mathematicians I have talked to); maybe they could learn the techniques, but the philosophy (of being able to prove anything, in principle) is genuinely difficult for some.

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    $\begingroup$ I'm finding the confidence point particularly true at the moment. I "learnt" engineering maths a few decades ago but really, I want to be able to do maths for its own sake and know that it's valid. So I've been trying to identify the chain of proofs all the way back to definitions and axioms for anything I use. This is giving me (i) hugely increased confidence in using the steps I now know how to prive, and (ii) greater confidence in writing proofs, as a result of practising by proving basics. $\endgroup$ – timtfj Jan 14 at 16:35

Disclaimer: I'm only really at the beginning of teaching myself analysis.

BUT I have a copy of Gelbaum & Olmsted, Counterexamples in Analysis which contains such examples as "Two uniformly continuous functions whose product is not uniformly continuous" ($x$ and $\sin x$), "A convergent series with a divergent rearrangement" (any conditionally convergent series, with a demonstration of how to get any predetermined limit as well), "A power series convergent at only one point", etc. (There is a whole chapter of power series examples.)

I believe there are other books in the series, covering different areas of mathematics.

Maybe books like this could be a useful resource? I didn't see a counterexample specific to your example, but I wouldn't be surprised if some of them might come in handy. And they're nicely collected together by subject, which might help towards being pre-armed with examples of what can go wrong.


(I know this is not an answer, but for readibility reasons, I'm putting this comment as an answer)

Am I reading your question correctly?
So, the student is saying:

$\sum_{n=1}^{\infty}a_n$ converges

$\sum_{n=1}^{\infty}b_n$ diverges to $+\infty$

Hence, the conclusion of the student is:

$\sum_{n=1}^{\infty}(a_n + b_n)$ diverges to $+\infty$

And your reaction is: this is wrong.

In my opinion, this is right. Can you give me an example of why this is wrong?

  • $\begingroup$ I believe the issue is that for the instructor, this is obviously wrong because the student hasn't proven/disproven the claim using strict definitions, like one usually does in analysis. Instead, the student has used an argument that is true most of the time without considering these tiny corner cases where his argument is not actually true. (Similar to saying, "I have a continuous function, so let me take the derivative of it" without realizing that continuous doesn't always imply differentiable, such as absolute value not being differentiable at $0$. $\endgroup$ – ruferd Dec 7 '18 at 13:17
  • $\begingroup$ Perhaps that splitting up the sum is not justified unless proven correct for the given sequences. $\endgroup$ – Jasper Dec 7 '18 at 13:21
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    $\begingroup$ I think you've rephrased what I said that the student said. I deliberately wrote that the student used the displayed equation in my question. So, asked to show that $\sum_n (a_n + b_n)$ diverges and you start by saying: "First write $\sum_{n=1}^{\infty}(a_n + b_n) = \sum_{n=1}^{\infty}a_n + \sum_{n=1}^{\infty}b_n$... and then...."etc. The first step is not OK because you've re-ordered an infinite series without knowing anything about its convergence (yet). $\endgroup$ – T_M Dec 7 '18 at 17:03
  • $\begingroup$ I think the student is basically saying "$\sum a_n$ converges to $A$, $\sum b_n$ diverges to $\infty$, therefore $\sum (a_n+b_n)$ diverges to $A+\infty$". $\endgroup$ – timtfj Jan 14 at 16:43

It sounds like the students may be treating sums with $\infty$ in them as just as easy to manipulate as the finite algebraic versions - presumably not just in this particular example, right? Which is very unsurprising, since a lot of times we do treat them this way and because they "look" like things you can treat wholly algebraically. (To really confuse them, spend a week on generating functions!)

More seriously, here is a possible thing to try if this happens in class: Can they write the (in general wrong) equation

$$ \sum_{n=1}^{\infty}(a_n + b_n) = \sum_{n=1}^{\infty}a_n + \sum_{n=1}^{\infty}b_n $$

in terms of the definition of the infinite sum/series? This is a good in-class "ask not a specific student, but everyone" question - restate this step in terms of its definition.

Once you do that (and if nobody can, then it's time to review it slowly), things look messier and presumably less "obvious" to the student(s). Now we are saying "limit = limit + limit". "Okay, can anyone think of a case back in calculus where the limit of a sum doesn't equal the sum of the limits?" Hopefully someone comes up with a situation like $\frac{1}{x}+\frac{-1}{x}=0$ at $x=0$.

Now you can say "in this case, we have to be careful too", and hopefully this analogy from something they should recall from freshman calculus can get you to convince them of it. And note that we are not asking them to remember things about series, which presumably they are still gnawing on, but function limits. (Hopefully someone will think of connecting the calculus example to the series with $\sum \frac{1}{n}+\sum \frac{-1}{n}\neq \sum 0$, but that will depend on the class.)

As a bonus, presumably you have a nice theorem about when the equation is true (such as, if both sums converge or whatever) which, when applied, would make their proof complete. Then on an exam you can give one where they are allowed to use that theorem, and one where they are not (do it "from the definitions").

I admit this is just one possible idea, but for some students it might snap them out of the blind algebraic manipulation, which I believe lies at the heart of many challenges in teaching real analysis. Analysis is really topology and the Archimedean axiom applied to the real line, but it still looks as algebraic as $(x^2)'=2x$ in many texts because of the notation coming (historically) before the proofs.


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