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I am teaching introductory real analysis this term and realize that my students have problem coming up with sequence in some arguments in real analysis. Let's take this example:

Theorem: Given a function $ f: [a,b] \to \mathbb R$ and $x_0\in [a,b]$. If for all sequence $\{a_n\}_{n=1}^\infty$ in $[a,b]\setminus \{x_0\}$ which converges to $x$, $\{f(a_n)\}_{n=1}^\infty$ converges to $L$. Then $f$ has a limit $L$ at $x$.

Proof Assume the contrary that $f$ does not have limit $L$ at $x_0$. Then there is $\epsilon_0 >0$ such that for all $\delta>0$, there is $x\in [a,b]\setminus\{ x_0\}$ so that $|x-x_0|<\delta$ and $|f(x) - L|\ge \epsilon_0$.

Then the next step is to choose (e.g.) $\delta = 1/n$ and come up with a sequence $\{x_n\}_{n=1}^\infty$ with $|x_n - x_0|<1/n$....

This step involves the (countable) Axiom of choice. Every time I perform a similar argument in class, they seem to understand it. But they are failing in the HW/midterm. It seems that their complaint is that they cannot see how to choose the sequence.

It seems to me that their confusion is legit, since this is the major reason why the Axiom of choice got some criticisms.

I would just throw "Hey! This is Axiom of Choice!" to them, but (1) this is not how we study real analysis here, where they don't have a solid background in set theory, and (2) that does not seem to help them understand the concept.

So my question is, how do we in general motivate the (implicit) use of AC in real analysis?

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Arctic Char
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    I have noticed similar issues with my students, but I suspect this has much more to do with abstraction in general than the axiom of choice, in particular. The idea that you can declare something to exist without having a specific example of it ... that's a strange notion for students to deal with. It may help you to think about it in these terms, and to explicitly point out to students that that's what we're doing in that proof. – Brendan W. Sullivan Apr 10 '19 at 20:29
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    Unless I"m overlooking some aspect of your example (some specific information about $f$), the sequence $(x_0+\frac1{2n})$ might not work. A function that doesn't have limit $L$ at $x_0$ might nevertheless have value $L$ at those specific points and oscillate wildly between them. – Andreas Blass Apr 10 '19 at 23:51
  • Yes, @andrea, by bad. So it seems there is no other way. – Arctic Char Apr 11 '19 at 00:07
  • I'm assuming @AndreasBlass would know exactly what logical prereqs are necessary here - is this really a choice issue? I read Stillwell's new book but don't recall "countable choice" coming up as an explicit axiom - maybe it's implicit in using recursion and/or various König lemmata, not that I understand all of that too well. And I'm sure the constructivists still can teach real analysis with similar issues in pedagogy. – kcrisman Apr 11 '19 at 01:27
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    @kcrisman This is indeed a choice issue. Zermelo-Fraenkel set theory without choice does not prove the theorem quoted in the question. In fact, Cohen's original model for the negation of the axiom of choice provides a counterexample. The theorem can be proved from a weak version of choice, namely choice from countably many sets of real numbers. – Andreas Blass Apr 11 '19 at 01:55
  • I have never quite understood the difficulty with AC. You have several, non-empty bags of marbles. AC tells us that it is possible to pick one marble from each bag. It doesn't tell you how to pick those marbles, just that it is possible to do so. That is all you need in some set-theoretic proofs. What is the problem here? – Dan Christensen Apr 11 '19 at 02:02
  • @AndreasBlass I didn't expect ZF to prove it, but I wasn't sure of the relationship between the different versions of choice. So full AOC is not necessary ... but what is (if any) the connection to things like WKL/ACA, or are those orthogonal issues? Sorry for the ignorance - I find these things interesting but very subtle, which I suppose is the reason why people had to prove these (non-)equivalences in the first place. – kcrisman Apr 11 '19 at 02:15
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    @kcrisman Even the strongest of the "big five" axiom systems of reverse mathematics, $\Pi^1_1\text{-CA}_0$, is provable in ZF (without any choice), so it won't give the theorem in the question. – Andreas Blass Apr 11 '19 at 02:19
  • Huh, interesting! I thought WKL was a choice axiom of some type, but I guess much weaker than countable choice? Thanks. – kcrisman Apr 11 '19 at 02:45
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    @DanChristensen I don't think people object to the marbles or socks/shoes analogy so much as that some of the consequences (e.g. Banach-Tarski) are more unsettling to some people. (I knew a guy who dropped the math major at that point and settled on philosophy as something more relevant to the real world and concrete.) – kcrisman Apr 11 '19 at 02:47
  • @kcrisman Philosophy is "more relevant" to the real world??? Hmmm.... Anyway, based on other comments here, I don't get the impression that Banach-Tarski is what is bothering students. – Dan Christensen Apr 11 '19 at 03:03
  • @DanChristensen just the messenger with that story :) no, the pedagogical issue here is unrelated to choice, in my opinion. – kcrisman Apr 11 '19 at 03:15
  • As some others have suggested (Brendan W. Sullivan in a comment, kcrisman in an answer), I suspect the major difficulty is the barrage of logically-quantified and symbolically-heavy expressions. For what it's worth, my recollection from the 1970s is that students tended to have MUCH more difficulty with (i) abstractly picking an element from a not-necessarily explicitly defined set (allowed by AC) than with (ii) making infinitely many such choices. I wonder if the huge increase in explicitness caused by coding and computer science exposure causes more beginners to have concerns with (ii) now? – Dave L Renfro Apr 11 '19 at 11:24
  • I was up against the character limit in my previous comment. When I was talking about "concerns with (ii) now", what I meant is concerns with (ii) AFTER being told that (i) is acceptable in mathematics. – Dave L Renfro Apr 11 '19 at 11:27
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    I originally wrote an answer in which I said (agreeing with the OP) that AC wasn't relevant, and then went on to make the same pedagogical points as in the other answers. Andreas Blass's comments have convinced me that I was wrong on the mathematical substance, so I deleted my answer. Yet I feel that it can't be true that AC is necessary in order to do classical analysis. One question is what definition of convergence we're using -- the premise of the OP's theorem seems like a good alternative definition of convergence. Doesr any well-motivated application requires theorems like this? –  Apr 12 '19 at 03:40
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    The following will be of interest to some here: Continuity and the Axiom of Choice AND Examples of Axiom of Choice used in introductory-level undergradute math AND Why is Axiom of Choice required for the proof of countable union of countable sets is countable? AND Choice vs. countable choice. – Dave L Renfro Apr 12 '19 at 04:31

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As others who answered have pointed out, this issue is not the countable choice involved in defining the sequence that makes this challenging for learners. Rather, the difficulty is the semantic complexity of the negation of the statement to be proved. When I get close to this theorem or similar ones when teaching analysis, I like to give a "fun" assignment a couple of days in advance:

Your friend asserts that for every $\epsilon$lephant, there is a $\delta$ay such that if the day is rainy, then the elephant forgets to bathe. How would you prove your friend incorrect?

Note the logical structure of the elephant sentence is very similar to the definition of a limit.

I am always surprised at the variation in incorrect answers, the rarity of correct answers, and the cognitive challenge this poses. It is revealing to listen to students discuss this challenge amongst themselves.

user52817
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I agree that the AOC is a red herring; that is not what the students find challenging here. My suggestion (and it is only a suggestion) is to consider taking a certain portion of your course and making it more "inquiry-based".

This is a bigger topic than can be adequately addressed in this space, obviously, but I have found that even quite weak students can really "get" at least some piece of truly difficult arguments, whereas when I've taught more traditional real analysis they seem to only partly get everything. For instance, you might have the expert in the Dirichlet function and where it is useful, or the expert in showing things are continuous, etc. (The best students will be expert in everything.)

For possible resources you may wish to peruse the following (disclosure; I've been affiliated with some of these by publishing or editing):

Real analysis is one of the more popular topics to teach this way. Naturally, you aren't going to "get as far" and it's not some kind of panacea that makes students magically get these arguments. Your mileage may vary. But I have found that, properly done, it can help some students who would otherwise always be lost understand at least one type of analytic argument fully, and sometimes helps the best students really know what is going on topologically and not just know how to parrot proofs.

kcrisman
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Like others, I don't think AC is the real issue. I don't think most students mean 'how can we do this infinitely many times?', but rather 'I don't know how to work out what to do'.

Personally, I would use the idea of 'what information do we have available to us?' I (in the role of a student) don't have any ideas for creating a sequence, but instead of giving up I should just play around with anything I can do, and see if that gives me new ideas.

What I have at my disposal is one definition I know to be true (the non-existence of the limit). Depending on where your students are up to (it sounds like they must be very strong students), that could take a few steps. If dealing with the abstract statement is too hard to comprehend, try it for specific values. $\epsilon_0$ is outside of my control (we could get away with pretending it is $1$), but I get to choose a small $\delta$. Choosing $1/2$ is a reasonable first step. I get handed back an $x$. Then try some other small values (some students might pick the sequence $1/n$, others $2^{-n}$). Each try hands me an $x$. Now making a sequence out of these seems less strange.

Jessica B
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