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I'm a third year student majoring in Math. Whenever I sit down and try to prove something, I just don't know what and where to start with. The first proofs course I took was graded very strictly so missing a very tiny detail made me lose a lot of marks (which does make sense since it is an introductory class to proofs and the "little details" could have been not "little").

But after that, I just get way too anxious when I do proofs because I don't know what kind of detail I would be missing. I end up completing the proofs by getting a lot of hints on where to start, and it takes way too much time for me to do a single proof (almost 2-3 days per one theorem). And because I don't want to get the proofs wrong, I keep searching up resources to do the proofs; so I kind of end up not doing the proofs myself. But when I see the "solutions" to the proofs, I realize they were very simple and I have been over-complicating it a lot.

I really love math and I want to be able to really understand courses like Real Analysis, and how scared I am with proofs definitely is an issue that I want to overcome. So my question is

(i) If you have gone through this stage, how did you overcome?

(ii) Are there any general tips on starting proofs?

Thanks.

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    $\begingroup$ I would talk to your tutor about how you might write your proofs better. Get them to read a proof of yours. They'll tell you which details you should highlight and which you could safely omit. You can also ask solution-verification questions here, but it's possible that we will have different tastes to whomever is marking your work. It takes time build this kind of intuition about proof-writing. Stick with it! $\endgroup$ Oct 6, 2022 at 6:37
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    $\begingroup$ What I recommend - since I am in the process of doing this myself - is to spend some time going over your old notes. Hopefully, you took notes of lots of the proofs that were covered in class: from them, it is good to highlight the essential themes of the proof. You will pick up / be reminded of the proof techniques. It is also good to have an intuitive picture of what you think the argument should be before you start writing. I don’t mean to memorise all my old proof-notes, just to refresh the general principles and techniques. And there’s also no shame in doing exercises from earlier courses $\endgroup$
    – FShrike
    Oct 6, 2022 at 6:38
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    $\begingroup$ I recommend my students to draw sketches regarding a problem. A drawing is never proof, but it can help you get a better idea of what is going on and more importantly, what it is you need to prove. $\endgroup$
    – AlvinL
    Oct 6, 2022 at 6:54
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    $\begingroup$ I don't have time right now to write a full answer, but this is very similar to what I used when I took MAT135 back in 2010. In short, apart from a proof without words, this course taught me when to use a proof by contrapositive, contradiction, or by induction by first going through logical analysis. $\endgroup$ Oct 6, 2022 at 7:58
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    $\begingroup$ I had the reverse problem as an undergrad. I was too impressed by slick arguments, I never learned to roll up my sleeves and give an argument which was completely correct, leading to math.stackexchange.com/questions/1283974/… . The way I got out of my problem was to mimic the proofs I read in books. Read them, copy them verbatim, then stew on them slowly, then try to reproduce them in my own words. $\endgroup$ Oct 6, 2022 at 20:42

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I also have an extremely bad issue with this, one thing that has helped me get through this feeling are this (non-exhaustive) list of things I have to remind myself whenever I take new course, or encounter a new subject, or exist in a mathy-space:

  • The only best way to learn, and get improve, is to start. Math is a learned skill, and like any learned skill you will suck at it at first, but you’ll improve as you keep practicing and attempting more to solve and write out more problems. Compare proof writing to other learned skills like painting, singing, etc.

  • Math is an incredibly social field, where networking with other mathematicians is a necessary thing to partake in to improve, and proof writing is a written social form of this. Essay writing is exactly similar, and as such it’s okay to have your peers proof-check your proofs and give you insights on how to improve. However, this itself is a big hurdle (at least for me anyways) because you will feel vulnerable, but that’s where you’ll learn. Find a community of learners like you and study and improve with them. Most professors would be ecstatic if you presented a problem to them and asked the to double-check it for you, because that shows an initiative to learn and willingness to hear ways to improve.

  • A perspective in essay writing that also plays its hand in proof writing is this: write everything out, with every little detail, and then go back through and cut out the fat, and make it look like you knew the entire time. You’ll miss tidbits here and there, and you’ll notice you over-explained parts there, but in that review process you’ll figure it out and get better.

  • Learning isn’t linear, if it takes two to three days to complete a problem or fully write out a proof, that is okay! One thing you must remember is that even if it’s baby steps, if your doing the work and effort to improve and practice you’ll get there.

The fundamental baseline is: practice, practice, practice, and it’s perfectly okay to mess up because that’s where you truly learn! With every mistake you learn a one new way to not do that route again, and with every mistake you catch you get better at not doing those same mistakes in the future. Proof writing is full of them, and even the best mess up time to time. If you browse the higher level questions on this site, you’ll see members with more than 100k rep who have been at this for years admit their answers or questions were wrong, and point out their mistakes and how they improved it. To find those, and feel reassured, look in the comments of most posts!

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Are you sure that you are understanding the proofs you are working with? In my experience, there isn't that much scope for genuine overcomplication in undergraduate mathematics. Instead, some students employ a scattergun approach where they mention a lot of things they consider true and potentially relevant, and hope that all steps of an actual proof are somewhere in there.

For the proofs you are writing yourself, make sure that you have clear picture of the purpose of each part. You may even consider drawing diagrams here: For each point that you make, mark both its prequisites and where you use it in the end. Do the same for the sample solutions that you find. Any part that doesn't actually contribute to the desired conclusion shouldn't really be there.

A more advanced exercise with a similar goal is to try and attack your proofs. If you can make yourself doubt certain parts, those are the parts that may need more detail to clarify why those doubts are not actually merited. A structured way to get there is to make small modifications to the statement you are trying to prove that yield something false. Which part of your proof breaks down?

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  • $\begingroup$ Yea a lot of my proofs initially were me just writing down like 50 things i know to be true and then assembling my result and then removing 49 things that ultimately were close to the topic but not related. Writing the proof and knowing what the proof requires are different stages and a lot of students try to do both at once. I later realized you can plan the proof as a sketch (and then identify the core tidbits you need) so the writing stage doesn't suck as much $\endgroup$ Oct 6, 2022 at 23:13
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In a way, you have already encountered the answer (or at least, a substantial component of it) and have alluded to it in your question:

But when I see the "solutions" to the proofs, I realize they were very simple and I have been over-complicating it.

Specifically, learning how to write proofs is an exercise in learning how "good" proofs are written; and as is the case with many different subjects, we learn by example. Exposure to how other mathematicians write proofs is not just teaching you the ideas in those proofs, but also their presentation, communication style, extent of rigor, and depth of elucidation.

That said, even in the realm of "good" proofs (where the definition of "good" is perhaps best left to the broad consensus of the mathematical community), there is quite a lot of variation in how much detail is provided. Some mathematicians tend to conflate brevity and elegance, claiming that a particularly terse proof is, by construction, a good one. While this may sometimes be the case, I do not think one necessarily follows from the other.

What matters for you, however, is the exposure: reading and seeing and thinking about many different kinds of proof, from different sources and areas of mathematics. Through this exposure, one develops their own style and gains experience with proof writing.

There are other ways in which we practice good proof writing, which is a subset of communicating mathematics in general. For instance, solving problems, and presenting their solution in a clear manner, is also an important component. Oral, in-person presentations or interactions are also very helpful.

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You should actually over-complicate everything if you like.
Also make mistakes.
Also write nonsense.
This should not be a taboo.
Then leave the arguments you have for 15 minutes (and watch tv for example) and come back and say. " How this could be expressed in a simple way?" "Is this actually correct?". And then just try again.
As you will see this is a simple, short answer which will work.

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I think the resolution is to have a very good understanding of the theory. Sometimes we try to do problems without giving time for theory to bloom in our heads. Alexandar Grothendieck once said, one should not prove anything which is not trivial.

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First time answer, please bear with me.

  1. Consider the logic of the theorem. a) Is it if A, then B (A => B) or A if and only if B (A <=> B)? In the second case, the general procedure is to prove A => B and then B => A. Usually, there is an easy direction and a harder direction. b) To prove A => B, you can either i) assume A, then eventually reach B; (ii) assume ~B (the logical opposite of B) and find ~A (the logical opposite of A); (iii) proceed with a proof by contradiction: assume that A is true and B is false, then eventually reach a contradiction (something impossible), then it means that B must be true; proof by induction: to show that a statement is true for all n>=1, show case n=1 and show that case n implies case n+1. There is some pattern recognition involved to build an intuition on which approach should be used.

  2. Divide-and-conquer: break the theorem into smaller blocks, then proceed to prove each block individually. For example, to prove A=>B, we prove A=>C, then C=>D, then D=>B where A, B, C and D are different propositions. To fit the blocks, think like this: "if I had X, then I would be able to prove X=>B", then prove that A=>X. Or divide the problem into mutually exclusive cases and prove each case one-by-one.

  3. Develop an intuition by working on small examples or by trying to find counter-examples. Start proving a simpler statement (say for a more restrictive class of function or in one dimension only), then see how it can be generalized.

  4. If you need to prove equalities (x=y), prove x<=y, then y<=x.

  5. Once the theorem is proved. Revise the steps and see if any simplification can be made. Check also for errors, with particular attention to signs and arithmetic error. Make sure that every operation is justified/legal.

  6. Read other proofs and learn what level of details is expected (proofs in papers/books might skip a lot more details than what is expected in class, but there is a bit of subjectivity in determining if you understood how to go from one step to the other when details are omitted).

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Lots of great suggestions here, especially working with other people. But here's one thing I did through most of college which is a bit extreme but really helps with these skills. I would solve homework problems on a whiteboard, and when I had solved a problem I would look at it until I thought I understood it fully, and then I would erase the board. Then the next day I would write up the solution. If my solution was a huge mess then there's no way I would remember it the next morning, but if I understood the ideas then it's usually pretty easy to reconstruct the argument.

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Here are some thoughts.

Logic / Clarity. I hear a lot about this perception that proofs require hyperspecialized mathese to write and interpret. While there are some special conventions about how words are used in math that are different from outside math (plus math terms themselves which do have technical definitions), my experience is that more often issues stem from failing either to understand pure logic or to simply be clear in normal-English ways.

By logic I mean propositions using terms and phrases like "for all," "there exists," "implies," "if and only if," "if / then," "and," "or," etc. and the ways such propositions can be created, combined, broken up, reshaped, and connected with each other in valid (truth-preserving) ways. If you're asked to prove X implies Y, and you start out your proof by assuming Y is true, you're simply not being logical, and that may have nothing to do with your understanding of X and Y themselves.

Plenty of errors in exposition and communication I see in proofs are actually not mathematical but linguistic, and by replacing math terms with analogous everyday terms (if possible) we can see how they basically translate to broken English, even by native English speakers who think they understand all the math terms they're using! Sometimes the issue is just not communicating enough: if you have a bunch of symbols moving around, you can off to the side explain what's going on with the symbols. If you're using certain theorems from the text or class or using the result of a previous result in a homework, or whatever, make sure you're explicit about this.

Big Picture / Small Details. Proofs involve lots of little details. If you use too little, people have too many blanks to fill in. If you use too many, unless you bake them into layers, people will get bogged down and find it hard to follow along. It's nice to organize your proof - often in explicit, visual ways - to make the big picture elements pop out the most and be easy to navigate with the naked eye, with details then attached.

A good way to do it is to chop the proof up into individual parts, or signposts. You can summarize ahead of time what's going to happen in your proof by citing the major signposts (often in papers, we call these lemmas), or you can conclude a proof with such a summary. When most mathematicians read papers, they skim through and identify the "big picture" ideas first, and then comb through the details second.

In written text I personally make heavy use of big vs. small text, underlining/bolding/boxing, left/right/center-justifying, arrows between things, etc.

Story and Meaning. Good proofs tell stories, where many quantities, expressions, and operations can be interpreted with some kind of meaning. Often proofs will choose to define or choose unexpected things, and go in all sorts of directions, and what makes this manageable as a reader is when you're told the thought process behind decisions, and given descriptions of what things are or what we're doing, "morally" speaking.

(As you practice this, you will build mental models of things in math that aren't easy to find in textbooks or told in many lectures. I think this is an unspoken psychological advantage many mathematicians develop, which disheartens outsiders who don't look farther than calling it unattainable, inscrutable genius.)

Apply Theory of Mind. There are many experiments in developmental psychology which investigate how children develop a "theory of mind," or an ability to conceive of others' minds as having their own beliefs, intentions, perceptions, etc. Neurotypical adults don't make the same kind of theory-of-mind mistakes we make in childhood, but we all have lapses, and it is a constant struggle to overcome this in any form of writing, and proof-writing is no different.

Consider what your proof looks like to someone who hasn't been thinking all of the thoughts you've been thinking up to writing it. If you start using a new letter out of nowhere, what will they think? They'll be confused. Sometimes we create blindspots in our writing that we consciously glaze over, so it can be good to leave proofs along and come back to them later.

Proofs by mathematicians for mathematicians are written and read differently than proofs by students for graders. What do you think goes through someone's mind when grading a proof? They probably have their own understanding of key ideas and will skim for them. Whenever you reason one thing follows from another, the grader will ask themselves, "is it possible someone could have misconception XYZ and still write what I see before me?" and even if you didn't have misconceptions while writing, the answer to that question may surprise you. Avoid being saying things that could be interpreted in other ways. (Sometimes the proofs I read give off the vibe of deliberate obfuscation to hide shortcomings. If so, I don't necessarily blame students for the hustle, but it makes me wary.)

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I have been in your case, and I got through it since I started to admit that there are some proofs that are not natural (especially in Measure Theory)! I know some theorems can be proved just by the definitions but some theorems need a special technique to prove them.

One simple tips is that you try to think of "what you need to get the proofs done" and "what do you have". For example: $\lbrace x_n \rbrace_{n\in \mathbb{N}}$ is a sequence in $\mathbb{R}$. Prove that $\inf x_n = -\sup (-x_n)$.

So looking at first, I completely can not prove the equality directly. So I choose another, which is a little bit longer, but easier to be done.

I prove $\inf x_n \leq -\sup (-x_n)$ and $\inf x_n \geq -\sup (-x_n)$.

So first, I need to look at the definitions of $\inf$ and $\sup$. First, since $\inf$ is the largest lower bound, then $x_n \geq \inf x_n \Leftrightarrow -x_n \leq -\inf x_n $. Since $\sup$ is the lowest upper bound, then $\sup(-x_n) \leq -\inf x_n $. So $\inf x_n \leq -\sup (-x_n)$. Can you prove the other one? :)

There are some simple tips yet useful:

If you want to prove $a = b$, then you can try to prove $a \leq b$ and $a \geq b$.

If you want to prove set $A = B$, then you can try to prove $A \subseteq B$ and $B \subseteq A$. To prove $A \subseteq B$, we can let $x \in A$ arbitrary and prove $x $ is also in $B$.

If you want to prove two function $f = g$, you can let $x \in A$ arbitrary and prove $f(x) =g(x)$.

Ofcourse there are more tips, but those are the useful ones.

And remember that there is no shame in looking for the answer to the proofs. The main point is that you can understand the key idea, and can try to prove it again yourself.

Things take time. Good luck!

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  • $\begingroup$ Since $\sup x_n$ is the lowest upper bound of $x_n$, $\sup x_n \ge x_n \iff -\sup x_n \le -x_n$, but then we also have $-x_n \le \sup(-x_n) \iff x_n \ge -\sup(-x_n)$. Since $\inf x_n$ is the greatest lower bound of $x_n$, we must have $x_n \ge \inf x_n \ge -\sup(-x_n)$ as required....?!!!! $\endgroup$ Oct 6, 2022 at 16:28
  • $\begingroup$ sure ;)) I hope my comment helps! $\endgroup$ Oct 6, 2022 at 17:39
  • $\begingroup$ Yes it did 😊! Thank you so much. $\endgroup$ Oct 7, 2022 at 7:51
  • $\begingroup$ Gosh, I hate having to prove both $A \subset B$ and $A \supset B$ but it just cannot be avoided in many cases $\endgroup$
    – Abastro
    Oct 8, 2022 at 1:32
  • $\begingroup$ Usually there is a case that it’s easy to see but yeah I hate doing so too. $\endgroup$ Oct 8, 2022 at 6:09
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This answer is unscripted and somewhat of a ramble. I can definitely relate to your struggles in writing proofs. Almost every question on this site is absolutely beyond me, and even the ones I attempt to answer sometimes result in either a downvote or a comment saying, "you are using so-and-so theorem wrong," or something along those lines. My best answers come from years of experience in integration and finding a way to mix proofs with them, yet I still struggle with integration to this day.

Basically, proofs are really, really difficult for everyone. It's supposed to be that way.

(Side Story) Back in the spring of 2019, back when I was 19 years old, I took my first math proof class. The material was completely different from what I initially expected as I had used computational methods, like evaluating determinants, applying Lagrange Multipliers for constraints, etc. I was extremely good at that side of math and was even around the top 3-ish in my university classes. But when my homework came down to something like, "Prove that if $a \equiv b \pmod{n}$, then $a^2 \equiv ab \pmod{n}$, where $a,b \in \mathbb{Z}$ and $n \in \mathbb{N}$" the first sentence I would write was "Suppose that $a^2 \equiv ab \pmod{n}$ such that a and b are Z and n is N" or some nonsense like that. My professor would nitpick every little detail just like your professor. I would always be surprised when I would get a $0$ out of $5$ on each proof I wrote and wondered, "I'm showing a LOT of work, how come I'm not understanding this?" I didn't want to just give up, so I kept on trying and kept asking my friends for help, and they helped tremendously. Since they understood the material better than I could, I would copy their writing style and even copy their proof (which is cheating, do not do that) just to scrape by. Unfortunately, I got an F in the class (I think I had a 55% or something like that) and was forced to retake it. I thought about switching my major to some type of engineering since that field is more of applied math and physics, both of which I was good at, but I didn't want to just give up and wanted to be one of the few people who could graduate with one of the hardest degrees at my university, so I retook the class. The next semester, I found myself blazing through the homework with ease and ended up doing well in the class.

(i) This is corny, but having friends helps out a lot. In fact, I don't think I would have been able to graduate with a math degree if I wasn't constantly texting my friends, "How did you do this proof? Because I got so-and-so but how come we can suppose so-and-so?" Emailing my professors also helped me a lot. I used to constantly email my real analysis professor (to the point of annoyance), saying something along the lines of "Hi Dr. so-and-so, it's Cody from your 12:00 p.m. class. I'm having trouble with this Riemann Integrability proof. At first I ..." and would proceed to share an entire page worth of work. I would always show everything I tried to show my professor I'm part of the learning process, convincing him to provide me with some hints until I wrote a perfect solution, with nice formatting, grammar, etc. The same goes for always attending their office hours.

(ii) Obviously you should know that practice makes perfect no matter who you are. The best way to get good at proofs is to just get your hands dirty and start practicing right away. If you don't know where to start, just start writing something. An incomplete and incorrect draft is better than having nothing at all.

(Answer) It seems like your biggest issue is being scared and anxious to write a proof because your professor would nitpick every little detail. You have to get over that fear. Get over it by embracing the proofs head-on. You shouldn't be scared to start writing down what ideas you have to prove something. If a person is scared to lift weights, how is that person going to get more muscle? If a person is scared of heights and wants to overcome them, they have to push themselves by looking down at a high height and experiencing the feeling of falling, which is what skydivers do. Understand that your professor is grading harshly for a reason, and that is to prepare you and the entire class for the harder math classes like real/complex analysis, number theory, abstract algebra, graph theory, etc. The proof class you're taking is absolutely nothing compared to the intensity of the higher-level classes. Take this advice seriously. I learned this advice the hard way and kept embracing my difficult math classes head-on. Even after failing over and over again, I ended up graduating last December. And now, I'm currently waiting to see if two of my American Mathematical Monthly solutions are correct. If they are, then I would be the first person from my university to solve an AMM problem since not even the professors could solve them (well, maybe they could if they spend a lot of time on them, but I digress). I even have another solution in mind for a different AMM problem (that I've been putting off to the side because of work and doctor appointments).

I hate to brag about that last part, but it's relevant to my experience of constantly failing math and how I was able to pick myself back up. The same can go for you, too.

Here's a textbook that helped me do well in my proofs class. I highly recommend reading it in your free time. It's pretty easy to read and it's written from the perspective of a struggling student.

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    $\begingroup$ "I used to constantly email my real analysis professor and would proceed to share an entire page worth of work" OMG XD $\endgroup$
    – Filippo
    Oct 6, 2022 at 17:07

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