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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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    $\begingroup$ 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. $\endgroup$
    – Brendan W. Sullivan
    Apr 10, 2019 at 20:29
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    $\begingroup$ 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. $\endgroup$
    – Andreas Blass
    Apr 10, 2019 at 23:51
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    $\begingroup$ @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. $\endgroup$
    – Andreas Blass
    Apr 11, 2019 at 1:55
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    $\begingroup$ @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. $\endgroup$
    – Andreas Blass
    Apr 11, 2019 at 2:19
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    $\begingroup$ @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.) $\endgroup$
    – kcrisman
    Apr 11, 2019 at 2:47

3 Answers 3

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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.

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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.

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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.

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