struggling with Real Analysis

[u/shashachuqin]

Currently a second year undergrad student taking real analysis. This is my first proof based math class and I have not taken discrete maths before either. I’ve already taken up to calc three and linear algebra, did pretty well in them.

The professor is using Understanding Analysis by Abbott, and the lectures mostly make sense to me. However, his problem sets are very very long and do not follow the textbook questions. He seems to come up with them from his brain…and he posts them every Saturday..which is also great.

I find myself really struggling to know what steps to take and where to go after reading a question asking to prove or show something. This makes me very worried for future exams.

Any and all advice would be really appreciated!

[u/Present-Doctor-8577]

ed reps. Start by writing down definitions and try the obvious thing even if it feels wrong

[u/Midwest-Dude]

With proofs, always, always, always write down what you are given, what the premises are. At the end of the page, write down what you are trying to prove. Use the definitions and theorems you've already been taught to get to the end. If you get stuck, see if there is a way to go from the end backwards - sometimes the two methods meet in the middle. Or, think of what you are missing that could be applied if you threw it into the mix.

If you get a problem you can't handle, sometimes moving onto the next problem and coming back later can help. Sleeping on it can also often open up new ways of solving the problem after you wake up. I've solved many a tough problem that way.

There are books on how to prove things, like Book of Proof, might help you. You don't need to read the whole thing.

[u/TwoOneTwos → u/Midwest-Dude]

If the only thing I know are definitions and I need to prove a theorem, I’ll write down every definition exactly as the textbook states and then just write what I’m thinking. Usually this helps me and I can figure out most if not all of the proof before hitting a brick wall and end up spending time working backwards

[u/shashachuqin → u/Midwest-Dude]

That’s very helpful thank you

[u/shashachuqin → u/Midwest-Dude]

Do you know how test for this kind of math course is usually formatted? I know it probably depends a lot on the professor, but I just don’t see how he can give hw problemset kinds of questions and expect us to finish everything in less an hour and fifteen minutes

[u/Midwest-Dude → u/shashachuqin]

I must say, I don't remember! That's what I get for getting old. 😐

Hopefully someone else can comment on this.

[u/TwoOneTwos]

Have you not taken an intro to math proofs class before…? I find it crazy how there’s no proofs course as a prerequisite for real analysis.

I’m probably going to get downvoted for this but this is just my opinion: Drop real analysis, take an introduction to math proofs course or at the very least pick up a book on proofs like “Book of Proofs” by Hammack.

[u/shashachuqin → u/TwoOneTwos]

No because it is not a requirement for my joint math Econ major. The only prerequisites for analysis is calculus and linear algebra

[u/shashachuqin → u/TwoOneTwos]

To be fair, my professor doesn’t expect us to write super thorough perfect proofs yet, I just have a hard time knowing what to do

[u/shashachuqin → u/TwoOneTwos]

I did also ask my TA the first week of class if I should drop this and take discrete first instead, he said analysis was also his first proof based math course and he thought it was fine, so I’m no sure

[u/ILoveLaggyInternet]

Thats the point. Have fun. Anything worth doing is hard.

[u/Fred_Scuttle]

I think that one of the challenges in analysis even for people who have taken other proof based courses is that you really need to forget everything you know about numbers and follow the axioms and definitions.

When I took my first analysis course, I showed my friends one of my first exams where a question was “prove 1>0”.  It’s funny that an advanced math student is being asked to prove something that everyone knows, but the point is that “1>0” is not one of the axioms or definitions that you are given, so can you show that it follows from them?

If your professor is good at his job, then I would expect that you would likely see exam questions where your intuition or analogy will lead you astray, forcing you to really think about rigorous proof

If you don’t already know some basic proof techniques you may want to look into getting  some resources on that.  If things like induction, or how to prove that a set S is a subet of T (assume x \in S and show that x \in T) or how to prove that two sets are equal (each is a subset of the other), you are going to need to learn that quickly as well.

In the end, though, in order to ace analysis, you really need to deeply understand these two things:

• If x is a non- negative real number and x< \epsilon for any \epsilon >0, then x=0.
• If sup(S) is not infinite and y<sup(S) then there is some x \\in S with x>y.

If you can prove them both and apply them, you will go far.

 

[u/Traditional-Month980]

I've got four pieces of advice. But TL;DR: real analysis homework is going to suck and it's good for you that it does, but there are common mistakes undergrads are prone to that you should avoid.

1)

I find myself really struggling to know what steps to take and where to go after reading a question asking to prove or show something.

Good. Embrace this feeling and try to feel it every day.

You know those feelings you would rather not feel, confusion, friction, stymied, etc.? Well when you're feeling these things, that's when you're actually learning and getting smarter. Successfully doing the problem is just "proof of work", in much the same way a PR in the gym is merely "proof of work" for what was surely weeks of struggle getting there.

2) The mistake most undergraduates make is giving up too soon and relying on solution manuals. If you struggle with a math problem (at this level) for ten minutes and give up, that's too soon. Two hours might be too long (depending on the problem). Thirty minutes to an hour of struggle before you look up a solution is better. Note: I'm not including the time it takes to write a "final draft" of your solution.

3) After reading a solution, wrestle for a few minutes with the questions "why would someone choose this approach? How did they think to do that?". It is okay if you cannot answer either question, sometimes it's just black magic fuckery. Then click away and try to recreate the solution yourself. It's easy to just read a solution and convince yourself you understand it on a deep level. Do not fall for this trap.

4) The humanities taught me this next one: thinking is writing, and writing is thinking. A thought is a nebulous thing of questionable value until it's given form and precisely stated. You won't know if an idea works, or even makes sense, until it's written down. So always be writing: take notes in lectures, work problems and examples while reading the textbook, try to construct your own examples when faced with a definition, and try to write complete solutions in paragraph form when you study. At some point before you graduate, take a writing class.

[u/bedrock_city]

You really need to find an introduction to mathematical proofs and work through the mechanics without also worrying about the analysis content. It's a whole new thing to start with a set of definitions and show how one fact implies another. You need some practice with basic truth tables, contrapositive, proof by contradiction, proof by induction. AI can probably help here, ask for a basic introduction with very simple proofs so you can start with the structure and then work up to more interesting mathematical content. 

Number theory is also probably a more intuitive first step once you have the proof idea down pat. E.g. a trivial thing like "prove that the sum of two even numbers is even" or "the product of two odd numbers is odd".

[u/Commercial_Mine_47]

1. Understand that for each topic we mostly begin with definitions.
   
2. From definitions then come the theorems. The theorems are usually if A then B kind of statements but there a few bidirectional theorems.
   
3. The definitions are usually used to prove the theorems. And “minor” theorems used to prove “major” theorems.
   
4. Don’t go beyond what a theorem says. For example, intermediate value theorem assumes continuity of the function. With that, try to look up counter examples - why the conclusion doesn’t hold if we remove continuity.
   
5. You need to memorize the major definitions and theorems.
   
6. Spend 2 hours after each class to review what was taught that day before the next class. This will help tremendously.

[u/StructuredChess]

Hard to give more specific advice without knowing the concrete problems you're struggling with. But yeah, higher level Math is supposed to be about using the tools you know to solve new problems, not about repeating the same algorithmic ones over and over again.

[u/Low_Breadfruit6744]

Depends on how hard your questions are. Some analysis class homework questions could be a few pages of writing once you figure out, you will just have to reset expectations that doing questions is a sprint.