Hey there. I am planning of taking a real analysis course but I'm so scared of the rigorous mathematical maturity involved in it. I am planning to get a leg up by looking at the material and seeing what I have to know. What are the main proofs techniques I can get a hang of in order to succeed in the course? I have only taken two proofs courses but I did kind of bad but want to get better.
Hi there, i find the book online. I haven't purchase yet; but before i do i want your opinion whether or not is it saved to take the course as independent study???? Thanks.....
@@youssephfofana9226 Analysis is a course worth taking from all possible sources. All your life. There's never can be "enough" of this only "true" mathematics.
Just took a real analysis exam with 20 proofs, 5 definitions, 10 T/F, 5 open-ended questions, and 3 T/F questions where you have to prove your answer. In an hour. Dropping out now. This was my last math course... I knew all the material, there just wasn't nearly enough time to write everything. Edit: I didn't drop out. After commenting this, I got angry because the prof said he didn't want me to pass so I studied hard and memorized every proof from 4 chapters of the textbook to do well on the final. Passed the course and I'm gonna graduate now.
@@ToddlerAnnihilator666 From what I've seen from some online math groups, there seems to be a stereotype of certain areas and their math faculty, I couldn't elaborate too much as I don't quite understand it or have the experience to speak on it's validity
@@soupy5890 Ok you are indeed correct. I was lucky to attend uni with fairly down to earth staff/professors. But those that transfered from other unis say that to keep their employment rates (and other statistics) higher they purposely fail students after certain qouta has been met
The video is quite helpful. Thanks for the suggestions. Struggling with the real analysis right now. Questions are not super hard, but I'm nowhere near solving them. It's frustrating that after so many hours of hard-working, I still have a hard time writing down proper proof without oversimplifying or complicating things. Whereas my classmates are gifted, discussing recondite ideas with prof all the time.
I'm actually kind of surprised to hear that (at least some) math programs use real analysis as the first proof-based course. At my school, they used the discrete math as the "intro to proof" course. As a computer science major, it was hard enough for me to pass this class when we were writing proofs about sets; I can't imagine having to throw calculus into the mix!
The more practice with mock exam questions, the more your confidence grows. I frequently visit Mathematics Stackoverflow and MathOverflow, considerably the best hubs where I view tough exam-like questions and others' solutions. I learn much by brain-picking gifted mathematicians there. Don't mind me, as I am only an intermediate in mathematics.
Agree on repeatedly failing and trying then finally getting it. That's probably the best thing that I gotten out of being a math major. There's no way I'm giving up on anything after completing my degree.
I totally feel you. The key here is really not giving up. I am taking analysis right now and it’s extremely difficult to me. But I still force myself to try again and again.
I came here after crying my eyes out because of a real analysis assignment. I really wish I knew this beforehand, it’s so disheartening to be in a class where you feel so inadequate
I will be teaching Real Analysis this semester. In preparation to the semester, I was planning to find a tasteful video on the history of RA. Then instead I found this video! These things came to me naturally when I was once a student, so, I was blind to most of the issues. But by experience I know, these are exactly where my students may struggle. I will share this video with my students. They'll surely appreciate it. Thanks for the excellent video!
Killer in what way? For context I'm considering taking Real Analysis as my elective (mainly out of curiosity and for the challenge) but I've heard often that Topology helps a lot with understanding real analysis and should be taken before analysis.
@@FsimulatorX If the presentation is just abstact and formal, then you might ask 'what is all this, and what are the practical examples that lead to these definitions?' For example, the Euclidean space R^n is a prototype for a toplogical space. A well motivated book is Topology now! by Messer".
Here's a nugget: Take Set Theory if it's offered or study it on your own. This is like the "introduction" to Real Analysis. This will get you thinking about the abstract logic when studying real numbers and knowing how to "speak the real analysis language" when it's time to discuss integration and differentiation utilizing sets because you're not going to be computing hardly anything, but writing nothing, but proofs and more proofs
My university does not offer a set theory course but it does offer an Intro to mathematical concepts/proofs course that covers logic, proofs, sets, functions, relations and number theory.
My God, what you just said here about the essential importance of knowing definitions really opens my eyes to so much, not just in math, but in life and the world in general...
2:38 induction definitions. alarm bells. I remember when I was sitting in an upper-division class and remember the panic when the professor would right Defn. - Now I understand. LOL 5:20 Start at what You know and The answer you want, meet the 2. 6:30 Notations, quantifiers, symbols,
I agree with you about not giving up and persisting. Sometimes the first encounter with some new material is like hitting an impenetrable wall. You have to persist in battering that wall, until it gives way to your understanding.
@@BriTheMathGuy When I first tried abstract algebra I saw the material and just about had a panic attack. I am not ready for it yet. I am brushing up on my fundamentals first before I try again.
You can do it! .. It IS hard to digest, but get yourself s many resources you can to understand how to deal with all the elements that make up the subject. Best wishes!
1) The real Analysis course is nothing like a Calculus course. It's logistically rigorous and proof-based. It's not taking derivatives, factoring, plugging, etc. 2) Be familiar with some proof techniques, such as mathematical induction. 3) Be extremely familiar with definitions: definition matters. You need definitions to start a logistic proof that your professor like. 4) Write down the definitions you know and write down what you want to prove (the conditions and the conclusions). It can be a great way to trigger a correct proof. 5) Be familiar with logistic quantifiers, all kinds of notations. 6) Persistence is key. You have to stick to learning and trust yourself. Never giving up is important to nail this class.
Yes, yes, his point number 4! Sometimes, just start "Since, (definition) and (definition), then because of .... we see that (conclusion.)" Write this in "math" to practice. Trust me, you will be on your way.
oh, I started real analysis book by myself just bc I found it interresting to learn something abstract myself PS: i am electrical engineer at high school. Nearly every question seems unsolvable while deffinitions are so simple and logic. I really felt stupid bc I couldnt proof such clear things like intersection with such simple deffinitions. But now I see that it's not that simpel as I though))))
Do you have any tips on how to (improve) remember(ing) math theorems and definitions? Most of the times I sort of get the point of how you can use it and understand what it does, yet I always struggle to memorize the more precise/small parts like for example some given requirements, preconditions or exceptions.
Real analysis was by far the most troubling undergrad math course I've taken as a math major. Don't know why I still have the textbook I don't understand 95% of the text lol
@@youssephfofana9226 I had a skinny little (80 pgs or so) book called Set Theory that was used in class. Sry can't remember the author but it might've been my prof (last name Bradley). Very concise explanations on axioms, (well) ordering, proof by induction, etc
One of the most difficult thing for me as a beginner is to use mathematical definitions to proof certain theorems...these theorems makes you realize that you don't know the actual meaning (essence) of the topic.. 😑😑
This subject really shook me in my first sem class. It takes so much time to absorb, and is just So different. Watching your video's making me realize how students first upon entering college-level math need to be given intro transition classes into each subject each semester. Like, u need some storytelling and constant context to get through that!
It means that for every epsilon greater than 0, there exists an n in the Natural Numbers such that for all n greater than some k (also a natural number), then the absolute value of the sequence (X_n) is less than epsilon. There!
I read and went through all the questions from Richard hammock's book of proof. I don't know if this is the same this as pure logic, but it is beautiful. To me, real analysis sounds like it builds on the bones of logic. But what kind of course or subject lies at a deeper more fundamental level. Like the three rules of thought and a proof for the existence of truth.
Being an undergraduate engineering student I didn't have an acquaintance with RA. I was really good at calculus, but RA kicked my butt when I got into grad school. I needed to strengthen my pure mathematics background for what I was into. And as you mentioned, it is NOT calculus. So I had trouble with the abstract nature of it. It still bugs me after all these years! And that is why I watched your video. Thanks.
Issue: What is a differential of an irrational argument? Let a= some rational approximation, and A be the irrational number itself (if that makes sense). Then A - a > dA and there is no way a + dA > A
Lol watching this one day after my real analysis exam has me like, wow....CAN DEFINITELY RELATE!!! By the way, you're the one who taught me what a topology is. Absolutely love your video on that, it was extremely helpful and I love this video. It's ℝeal lol. It's sorta sad the way the math program is made though, before this level, no one will know what's about to hit them. I only knew that everyone said they "hate" real analysis. Never heard nor saw anything else about it. Now I understand that it's an analysis based course, which I kinda like tbh, and kinda hate. I loveeeeee finding solutions to problems lol it makes me feel like a jackpot winner LOL, but on the other hand, I HATEEEEE all of these seemingly random, yet connected and absolutely necessary theorems and definitions. I stopped counting them when we hit topology. LOL. I do however have a very LONGGGGGG list of almost all of the definitions and theorems stockpiled into a MS Word document hahaha... Crazy stuff. Tbh, Advanced Calculus was much harder for me than ℝeal analysis because at least by the time I hit ℝeal analysis, I was already exposed to a great many of the definitions that we would need to use.
In my current undergrad experience its interesting the courses which definetly are built upon complicated niche rigorous proofs are often the classes where you arent required to do those proofs, because many students who find calculus appealing end up reaching into areas like differential geometry and geometric calculus those courses are often taught at the undergrad levels with out too many proofs even tho there “past” calculus they require real analysis to work
At 6:49, you mention taking an "introductory proof class to understand real analysis". Can you please recommend any book or materials that one can review? Thanks,
Currently doing a major in applied math and computer science and Real Analysis was one of the most dreaded courses. Unfortunately got stuck with Real Analysis, Complex Analysis and Stochastic Processes all in one semester haha. But who doesn't love a good challenge ;)
lol how did it go man? I'm thinking of taking real analysis, abstract algebra and stochastic processes on the same semester. Complex Analysis sounds dreadfully worse though XD
Interesting. I see how it would be difficult to unlearn Calculus from the geometric approach as it is introduced and developed in the usual way, then relearn it from an arithmetic/algbraic/logic point of view. *Me:* "I understand Calculus." *ANALYSIS:* "You don't understand the Number Line."
If people knew what lie in store for them, they would pay more attention in calculus to the proofs of the theorems, not just the applications. I remember I had the "I just want to know how to do this" mentality. As Brian mentions in his videos, you are in for a rude awakening if you do this once you get to real analysis.
Great video. Here's what you must know before becoming a math major. Real Analysis (aka 'Advanced Calculus') is usually the first proof-based course in college and it isn't normally taken until FAR TOO LATE to realize you don't understand proofs at all and you may NEVER be able to handle proof-based courses, making a math degree effectively unattainable for you. Real Analysis is usually taken in your junior year and these proof-based courses make up your last two years of college yet they are a COMPLETE CHANGE OF DIRECTION from what you thought math was all about. It's almost criminal that this isn't revealed and emphasized EARLY ON in college, but it is not. Math profs at university have pure math PhDs and were born knowing how to do this stuff (no kidding). They assume you are the same. College advising sucks and no one will tell you the following. Being good at 'plug 'n chug' will not help you with proofs. You can be an A+ student in all calculus courses and fail proof-based courses such as Real Analysis. If you happen to be reading this before it's too late, get Laura Alcock's book 'How to Think About Analysis' and STUDY IT CAREFULLY. If you can't follow her material, you are probably in the wrong major. Her presentation is, by far, the best I've seen. Even applied math majors will normally get a big dose of proof-based courses that may stop them - dead in their tracks. Get lots of help or change majors, NOW! Asking the prof for help is a waste of time. Get online and find someone on Wyzant or something like that who is good at pure math and do one-on-one tutoring. But, good luck! If you can't do proofs, a Math PhD is not going to happen. So, even if you complete an Applied Math program, you better have computer science skills in order to get a job in a relevant field (except teaching public school). You won't be teaching at University without a PhD in math.
@@xHannaHx33 What book are you using and what section are you lost on? If I can find the book online, I'll take a look at the section. Remember, get help ASAP. Don't wait. Perhaps try Wyzant. Upper level math is not commonly found on RU-vid. If this is your first 'proofs' class, it will make or break you.
All of this is why they should teach Real Analysis *before* Calculus. If you're balancing a checkbook, you don't need Calculus. However, if you're using Calculus, you need to understand the tools you're using.
It annoyed me into oblivion that in my math class I had to do "proofs" that weren't proofs - more like a basic vague sketch of ideas that really has no structure - at all and it was almost like the teachers were actively trying to persuade us that proofs don't exist. I can't even talk to my classmates now, because just a slight derail from the beaten path of plug and chugging leaves them confused, and that's kinda sad, being unable to really share these great ideas with anyone as an EECS undergrad(At least I hope that's the reason) :D It's a lot of work to learn this properly though
This is so true! I'm in my final year as an Electronics undergrad and in most of our classes that involve maths to some extent, the teachers really don't care about rigour and we just take most things for granted...
I hated that I didn't know proof as well as I needed, so I took a break, studied a HUGE amount of proof-related material. It's an advantage you have as an older student where you don't care how long it takes to graduate. My goal was simply to not complete a course/class unless I considered myself to have mastered the material.
I watched this video before starting the real analysis course so I couldn't understand what he was talking about back then. Now, 2 months later, I can understand and relate to each and everything he is saying. I followed what he told and I can say I am starting to get a grip, still very very very far from the end but I am moving atleast. Edit: I think I'm gonna fail this semester 😭
I really had a bad experience on Real Analysis. One time the Last Exam which was a week before finals got delayed to two days before finals exam. I attempted to cheat becasue It was really stressful studying for a long test and Finals Exam that was in the same week and it's the same subject. I attempted to cheat but got caught. My teacher gave me a chance trhough removals exam. As of now she still hasn't checked removals even when I took the exam on Feb. IM WAITING.
How would you be able to really cheat on a Real Analysis exam though? From my experience, even if you had the whole text book in your hand during the exam, the questions would still need such rigorous and complicated answers that in the end your book won't help you that much.
my test was like this. Each student had to bring their own pile of bond paper to write our answers since some proofs are long. i would bring lots and hide the cheat sheet within the pile. However during finals My teacher was the one who gave the bond paper and unlike the other exams where she stay put......she roamed around the room. It was hell. I couldnt do it. I confessed tp her about my attempt to cheat and she gave me another chance
@@TML-zw9dt Ah, interesting. I suspect she must've known that there would've been at least 1 or 2 people to bring cheat sheets. About the fact that she hasn't checked your removals exam, try to remind her about it as often as you can. No seriously, REMIND HER BEFORE IT'S TOO LATE. I was once sort of in the same position as you, but with a Chemistry class. So a few of us failed the class, but because it was a mandatory course and none of us were actually chemistry majors, the teacher decided to give us another chance by giving us lots of assignments. He said he was going to send the questions to our emails, but guess what? HE NEVER SENT THEM. I kept on texting him once a week to ask about it because my whole grade depended on it but HE NEVER REPLIED. I regret never actively seeking him out in person, but then again, that wasn't the only thing I had to deal with at the time. So at the end of the semester, I didn't get a C, a D, or even an E, but a big red I, as in Incomplete, because it was assumed that I didn't complete my work, and then had to retake the whole class again.
It’s funny how big RU-vid is. You are clearly doing a great job, you have videos that are well made and well targeted and raking in the views. This is the first video I’ve seen of yours. Keep up the great work!
just finished my first analysis 1 exam. had taken three rigorous proof based courses before this. taking abstract algebra at the same time. nothing could have prepared me for that 50 minute hell…
What's book do you use for introduction proof??? The name and the author precisely the ISBN# will be great. I'm looking forward as independent study. Thanks...
@@youssephfofana9226 hmm I don't have the ISBN, but it's called "Analysis with an Introduction to Proof" by Steven Lay. Here is the link: www.pearson.com/store/p/analysis-with-an-introduction-to-proof/P100001370585/9780321747471
Where I live it’s really different. The “Calculus” you learn in your first year is something in the lines of Spivak’s. A rigorous approach where you do both computations and proofs, but without introducing topological notions. That’s introduced in your second year. The same goes for nearly every subject you encounter. If I’d have to give my two cents though, I think both ways have their pros and cons. What I think it’s mandatory is a transition course that covers sets, maps, logic, proof techniques, etc. 1/4 out of 80 sophmores that take the math degree drop in the first year or earlier.
@@mateusvitor2774 yeah the rate of failing is quite high in the first year. The same applies to linear algebra, for example. It’s not in the realm of, say, Sheldon Axler’s LADR, but it is quite abstract for people coming from high school. For example, the last part of that course covers the dual space, which a significant leap in abstraction for people who are still coming to terms with vector spaces and linear transformations.
@@javiermd5835 agreed. I remember taking linear algebra during my second semestre. First exam was proof only, very different from, for example a calculus test during the first semestre.
@@mateusvitor2774 when I first took linear algebra literally half of the class was people who had already failed. I remember that in the first test only 33% of people passed. There were lots of really bad grades (in a scale from 1 to 10 most people were in the 1-3,5 ballpark). I know this situation is not as common in other countries, I’d like to know why because I don’t think people here are more stupid to be honest.
Real analysis is about the core nature of real numbers.I personally feel the reason why students find it difficult is that they study functions,limits ,differentiation ,integration and sequences intuitively. Most students do not have a sound knowledge of set theory and logic and therefore lack rigor in their veins.
Thank you Soo much dear.....your vedio is very helpful for me ....I was stuck .... totally stuck in maths Carrer due to ....real analysis....from Pakistan 🇵🇰.....I think maths like counting 1,2,3. +-× etc but in real analysis eplisone etc
I felt soo horrible watching this because I have to take real analysis as a physics undergrad and its soooooooooo awefully (insert adjective). I hate it and I'm scared of it, but I hate to go throught it...
I studied Measure Theory without any knowledge about Real Analysis. For 3 months up until now I studied every day, starting from the definitions of open sets, etc. I still remember the first several classes I could not understand what prof said. So I just quitted the class and studied it myself. Now I can understand the proofs and the solutions of exercises, but I feel like I’m still far away from solving exercises 😂. Tomorrow I have exam. Wish myself luck 😅
@@gauravbharwan6377 Yes! 16/20. I came back right after I had result. Turn out one (over 3) exam question was similar to one exercise and I learnt it by heart, so... well
Real Analysis as the FIRST proofs-based course? That's insane, I had my own introduction to proofs class and apparently talking to some computer science students my introductory class was three discrete math classes combined. Then I picked up a Real Analysis book to see what it's like and the first Unit just casually summarized everything I learned in that class. I'm still trying to read the book; it's intense and the exercises are brutal, but I find RA VERY rewarding. I was forced to switch Real Analysis with Abstract Algebra and Probability Proofs for the upcoming year, but hopefully I can take RA soon.
How was your 2nd Real Analysis class compared to your 1st one? I did really well in my first class (my school calls it Advanced Calculus 1), and I start the 2nd class of the sequence next week.
Thank you so much for the help. But I am curious to know the level of Calculus that Real Analysis is covering. Are the topics of Calculus IV included in this course? Do I need take Calculus IV before it?
An intro course in Real Analysis covers up to Rienman integration. So you don't really need much besides some level of mathematical maturity and single variable calculus
My first real analysis course didn't go beyond limits for calculus content. There was a big focus on sequences. My second real analysis did derivatives and integrals from R^n to R^m.
@@youssephfofana9226 I am sorry for not being able to help. But I encourage you to post your question on Quora. I hope you will get a good advice from the right people.
Seems like the US math Major is quite different from Germany (maybe Europe in general). Here you start with the real rigorous shit in the first Semester and it never stops. Calculus doesn’t even exist here lol
I study engineering and in my first year I have analysis twice, once each semester. It's pretty hard but we don't have to proof anything but rather calculate things. Maybe because I'm not studying maths. An example is we have to calculate double or triple integrals, convergence of sequences etc.
We have our real analysis in second year and it's just a commencement of the study ....and before choosing this subject I was really threatened by lots of my seniors tht it will just make u sweat from your teeth don't dare to choose it and here in your video I found the same vibe 😁🤣 anyways u didn't mean to discourage but I guess it's the fact ...and I liked the idea of your encouragement to never give up at the end♥️
Before taking the class, all this advice sounded obvious. After taking the class however, all this advice sounds painfully personal. Suffice to say i had to go over all the proof-based classes again after finishing the first year.
so glad i was able to take geometry and abstract algebra before RA. i had never seen non-E geo before then, so that solidified the importance of definitions. the set theory and structural concepts of ab. alg. helped me get my mind right. RA was still a challenge, but it’s like i was looking at calculus from “the other side,” so i really felt like i was going somewhere in my math understanding
