# What do you do when you are completely lost during a proof?

We've all been in the sort of situation where your professor is doing a proof/derivation, and the whole thing goes completely over your head (if you don't know what I meant, I envy you). And when they ask for questions at the end (or in the middle, even), no one else speaks up, and nobody wants to feel like the only confused one, so your questions go unasked.

I think it's also sometimes the case that your "question" would be "I have no idea what you're doing, or why", and that would feel a bit silly to say in class.

This has been happening to me a lot lately in my real analysis class; we have no assigned text, and my professor is very busy and has only two office hours during the week, so it's hard to ask them questions sometimes.

Anyway, in such a state of confusion, what is the best way (in your experience) to get a firm grasp on difficult proofs? Sometimes it seems that certain parts of proofs are pulled from nowhere, and I cannot imagine myself being clever enough to come up with such an argument, which deeply upsets me.

Perhaps some people would suggest I forget about the proof and accept the theorem as true, but this isn't a mathematical way of thinking to me.

• Make a coffee, step outside for a moment, turn back three pages or so to identify the thing that I must have missed without realizing. – Emily Feb 4 '15 at 21:54
• One thing you should be aware of is that real analysis took two centuries to really come together in any meaningful way. It was about 150 years after Newton and Leibniz that we even had a proper working definition of what a limit really is! Mathematics is not built in a day and it took very, very bright men a long time to come up with the tricks and techniques we use in courses and in research. It's not expected that you would necessarily think up all of the techniques yourself when the brightest men in the world struggled with it! – Cameron Williams Feb 4 '15 at 21:55
• Collaborate, find alternative resources and don't stress artificially about whether or not you could figure all of it out on your own without texts. Proofs are hard, even for us rather experienced mathematicians. It takes a long time to train yourself to think outside the box and develop your own techniques. Also.. don't try to compare yourself with the mathematicians that you read about in texts. It only sets you up for failure. It's easy to get caught in thoughts like "well with guys that smart around, why should I bother?" The answer is even us non-prodigies can be very good mathematicians. – Cameron Williams Feb 4 '15 at 21:57
• Also.. sometimes a certain proof just doesn't mesh with the way you think. It happens to all of us. In that case, maybe try to understand some of the pieces but don't worry about memorizing it $100\%$. No one person can memorize $100\%$ of the mathematics they come across. – Cameron Williams Feb 4 '15 at 21:59
• I'm always struggling. Taking a step back at looking at the pieces separately can help find a new perspective on things. – Karl Feb 4 '15 at 22:00

You can always ask for an intuitive explanation of a proof. For example, the proof that the real numbers are uncountable is actually quite easy if you understand the intuition behind the standard proof. Also the proof that an isometry $f:X \to X$ for a compact metric space $X$ must be surjective also has an intuitive explanation. (It must be "volume preserving" and hence onto because it's an isometry and thus distance preserving). There is often an intuitive explanation behind a proof, some key idea or something, and it can do a great deal toward you understanding the entire proof.