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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
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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
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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.

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(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?

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  • $\begingroup$ 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
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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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