Tips for Approaching Proofs and Theories in University Math Courses

[Kemilss]

Hey, I'm adult but going to university for the first time, and I am taking Physics. My first semester is all about Math. Algebra and Analysis.

I'm usually pretty good at math, but right now, we are "laying the foundation" for calculus, and of course all these things we took for granted over the years. This is proving to be an interesting challenge, which of course I suppose is good!.

Anyways, my question is, what tips do you have to approach "proofs" or "show that", using certain axioms? Did you struggle with these at first? How long did it take you to begin to be able to show these?

ex) let S be a nonempty subset of real numbers which is bounded below. Let -S denote the set of all real numbers -x, where x belongs to S. Prove that inf S exists and inf S = -sup(-S).

So to show myself what was being asked, I just made an arbitrary set, S, and found the inf of my set and compared it to the -sup(-S) of -S. Of course it was equal. But how should I approach "prooving" it!.

 

[homeomorphic]

I didn't have any difficulty with my first analysis class, so I wouldn't even really be aware of what the fuss is about with the difficulty of basic proofs, if it were not for what other people tell me, but part of that was that I came in well-prepared, having read Visual Complex Analysis, taken an easier proofs course, and studied subjects that improved my visual and logical thinking from engineering and computer science.

The first step is often just to convince yourself that it is true intuitively. You can just take an example to convince yourself, but you probably want to choose an example that is kind of general, so that you're not really using anything about the example you chose. The example is just there to help picture it better. I would draw a picture of -S, look at the sup and compare it to the inf of S. That's how I would start. Then, I would recall the definitions of sup and inf. Even write the definition out explicitly on paper if you have to. Then, I might work backwards from the definition and figure out what I'm trying to prove, and write that down. Then, break out the inequalities and go at it with them. With more practice, you might be able to do more visualizing than drawing and more thinking than writing, until the later stages of the proof. Usually, what I do is write a rough draft proof first, which might get a little messy, before I write the finished version.

Another thing is, the way I see it, the point of analysis isn't so much to lay the foundations of basic calculus. If it were that important to lay the foundations, just to be able to use calculus, physicists and engineers would all be studying real analysis. I don't think they are really "taking it for granted"--at least the people who think deeply about it. They just understand it non-rigorously. Having better foundations is just a bonus, in my opinion. The real point is to lay the foundations of more advanced topics like Fourier series and PDE (see the Discourse on Fourier Series by Lanczos or A Radical Approach to Real Analysis) or topology. Also, if you're like me, you grasp the basics of calculus a little better, just because you go over it a second time--sequences and series were very mysterious to me, until I took real analysis, but it's not the only way to do it. So, my point is, I think it's not good if people come out of a real analysis class thinking that there's nothing they can do now that they couldn't do before, aside from just prove things or, at best, reinforce the knowledge they had from calculus.