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What to do if you're about to fail real analysis for the second time?

A few months ago a Reddit user came to r/math with this tale of woe:

I failed real analysis horrifically the first time … and my resit takes place in a few days. I still feel completely and utterly unprepared. I can't do the homework questions and I can't do the practice papers. I'm really quite worried that I'll fail and have to resit the whole year (can't afford) or get kicked out of uni (fuck that).

Does anyone have tips or advice, or just any general words of comfort to help me through this mess? Cheers.

The first thing that came to mind for me was “wow, you're fucked”. There was a time in my life when I might have posted that reply but I'm a little more mature now and I know better.

I read some of the replies. The top answer was a link to a pirated copy of Aksoy and Khamsi's A Problem Book in Real Analysis. A little too late for that, I think. Hapless OP must re-sit the exam in a few days and can't do the homework questions or practice papers; the answer isn't simply “more practice”, because there isn't time.

The second-highest-voted reply was similar: “Pick up Stephen Abbott's Understanding Analysis”. Same. It's much too late for that.

The third reply was fatuous: “understand the proofs done in the textbook/lecture completely, since a lot of the techniques used to prove those statements you will probably need to use while doing problems”. <sarcasm>Yes, great advice, to pass the exam just understand the material completely, so simple, why doesn't everyone just do that?</sarcasm>

Here's one I especially despised:

Don't worry. Real analysis is a lot of work and you never have the time to understand everything there.

“Don't worry”! Don't worry about having to repeat the year? Don't worry about getting kicked out of university? I honestly think “wow, you're fucked” would be less damaging. In the book of notoriously ineffective problem-solving strategies, chapter 1 is titled “Pretend there is No Problem”.

This was amusing but unhelpful:

We are all going to die. Compared to death, real analysis is nothing to fear.

(However, another user disagreed: “Compared to real analysis, death is nothing to fear”.)

Some comments offered hints: focus on topology, try proof by contradiction. Too little, and much too late.

Most of the practical suggestions, in my opinion, were answering the wrong question. They all started from the premise that it would be possible for Hapless OP to pass the exam. I see no evidence that this was the case. If Hapless OP had showed up on Reddit having failed the midterm, or even a few weeks ahead of the final, that would be a very different situation. There would still have been time to turn things around. OP could get tutoring. They could go to office hours regularly. They could organize a study group. They could work hard with one of those books that the other replies mentioned. But with “a few days” left? Not a chance.

The problem to solve here isn't “how can OP pass the exam”. It's “how can OP deal with the inescapable fact that they cannot and will not pass the exam”.

Way downthread there was some advice (from user tipf) that was gloomy but which, unlike the rest of the comments, engaged the real issue, that Hapless OP wasn't going to pass the exam the following week:

I would seriously rethink getting a pure math major. It's not a very marketable major outside academia, and sinking a bunch of money into it (e.g. re-taking a whole year) is just not a good idea under almost any circumstance.

Pessimistic, yes, but unlike the other suggestions, it actually engages with Hapless OP's position as they described it (“have to resit the whole year (can't afford)”).

When we reframe the question as “how can OP deal the fact that they won't pass the exam”, some new paths become available for exploration. I suggested:

[OP] should go consult with the math department people immediately, today, explain that they are not prepared and ask what can be done. Perhaps there is no room for negotiation, but in that case OP would be no worse off than they are now. But there may be an administrative solution.

For example, just hypothetically, what if the math department administrative assistant said:

You can get special permission to re-sit the exam in three months time, if you can convince the Dean that you had special extenuating hardships.

This is not completely implausible, and if true might put Hapless OP in a much better position!

Now you might say “Dominus, you just made that up as an example, there is no reason to think it is actually true.” And you would be quite correct. But we could make up fifty of these, and the chances would pretty good that one of them was actually true. The key is to find out which one.

And OP can't find out what is available unless they go talk to someone in the math department. Certainly not by moping in their room reading Reddit. Every minute spent moping is time that could be better spent tracking down the Dean, or writing the letter, or filing the forms, or whatever might be required to improve the situation.

Along the same lines, I suggested:

perhaps the department already has a plan for what to do with people who can't pass real analysis. Maybe they will say something constructive like “many people in your situation change to a Statistics major” or something like that.

I don't know if Hapless OP would have been happy with a Statistics major; they didn't say. But again, the point is, there may be options that are more attractive than “get kicked out of uni”, and OP should go find out what they are.

The higher-level advice here, which I think is generally good, is that while asking on Reddit is quick and easy, it's not likely to produce anything of value. It's like looking for your lost wallet under the lamppost because the light is good. But it doesn't work; you need to go ask the question to the people who actually know what the solutions might be and who are in a position to actually do something about the problem.

[ Addendum 20220222: Still higher-level advice is: if you're losing the game, try instead playing the different game that is one level up. ]

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