The Best Way to Get Ready for Real Analysis #shorts

The Best Way to Get Ready for Real Analysis #shorts

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[sapientum]

Doing advanced mathematics without knowing proofs is like running a marathon without legs.

[leonardodossantosferreira]

I'm a physics undergrad and this semester I'm taking a course on Real Analysis. To me, the greatest difficult was to translate my intuition into real proofs. My tip would be: before trying to search for the answers to see if they match your attempts, you should try to do the same proof in every single way your intuition suggests you is possible, because doing so, beyond practicing, you train you creativity and learn how to structure your intuition as a mathematical proof. Definitly, this Real Analysis course is pushing me towards a graduation in mathemathics :)

[martinboites]

The best way I think it's to get some intuition of the concepts (the definitions) and lower your expectations by understanding how hard was even for those who were developing it: Cantor, Dedekind, Weierstrass, etc

[cosmicspectrum]

I see so many say to just get good at proofs and practice but I want to give more specific advice as someone who has taken Real Analysis. You should understand the ordered field axioms, the completeness property, and above all else understand how to do PROOFS INVOLVING INEQUALITIES. I put that last one in caps because that is a big one no one talks about and it took me a while to figure out. Most real analysis proofs involve dealing with inequalities. If you understand inequality proofs the triangle property makes so much more sense as to why its needed and used so much.

[bensherwin]

Book of Proof is a great book so far and definitely has taught me a lot about proof writing this semester. This is my first proof based math class (Sets and Logic) and I think the book has done a good job in prepping me for higher level math. I also have How to Prove It but haven’t read it yet.

[ishaangoud]

Same applies for any pure math class like Abstract Algebra, Topology, and Linear Algebra.

[edisondoko]

There is a book called “How to Prove It: A Structured Approach, 2nd edition” by David J Velleman that I really like.

[tmann]

I remember when I thought calculus was the hardest thing ever in high school but now I hear from the advanced mathematics side how hard other higher maths are haha! Is it bad that the more I hear how hard the subject is the more I want to take it? I like to shave against the gain and appreciate being a little masochist mentally. The sense of accomplishment and dopamine rush I get when I tackle hard endeavors is the best feeling ever. Even the little accomplishments I love too. Its like there is a point where you know that struggling is a side effect of success.

[青雲浮遊]

I have done 2 analysis classes. Next year is the real one or 3rd one. Here is my view:
1 Book of proof is really good for building the foundation if you don’t have one.
2 get use to ε-δ way of expressing mathematical ideas. It is used constantly.
3 be able to walk through each proof while establishing a big picture of theorems.
4 use examples to conceptualize and look up online for counter examples
5 more problems you solve, stronger you become. Mathematicians are unimaginably strong. So never stop levelling up.

[football]

Agreed! even though I didn’t take a real analysis class but proving limits using epsilon delta stuff is a tough thing

[gino_jihai]

Sir, recently I've borrowed a copy of "A Radical Approach to Real Analysis" by David Bressoud. I was positively surprised as the author does not organise the contents in a strict logical-sequential order, but according to the historical development of the topics, with emphasis on mistakes and improvements. For example, he shows how the solution to the Laplacian equation of the heat transfer problem of a planar bar solved by Fourier, i.e. an infinite cosine series, generated a "crisis" at that time.
The author believes that this is the way analysis should be taught as it would be more intuitive and stimulating. I do recommend it!

[kchannel]

I always had the impression you needed a firm bases in calculus before tackling Analysis. I guess I need to brush up on discrete since it has some good lessons on writing proofs.

[texasgoat]

Undergraduate 2nd time taking analysis, now at the graduate level. Only class I’ve gotten a B in! Tough class, but one where mathematicians truly developed...

[mpcc]

Study Elementary Number Theory, Naive Set Theory, Euclidean Geometry, Calculus, Linear Algebra, and Learn to write proofs.

[giu_mal]

The best way to survive in real analysis is starting to GET ALONG with the proofs. I like to emphasize it as UNDERSTANDING the proofs requires a good portion of endurance.

[l.w.paradis]

I'm not gifted in math, and am horrible at spatial intuition; I'm super-verbal, and proofs are easier for me than are most other things in math. I came to love math when I found a verbal way into understanding it. :)

I admire the "real" math people. :)

[zhangkevin]

Honestly can’t know how hard it is until you try, I’ve seen even CMO qualified people struggle on real analysis so you are definitely not alone. It’s just that one thing you have to experience to understand how hard it is.

[chinaechetam]

I’m a mathematics undergraduate and real analysis is one of the confusing things I’ve come across

[SequinBrain]

they should have a proof writing class prior to the classes that need proofs.

[randallmcgrath]

It is interesting that many non-math mahors assert that Number Theory, Topology, Real Analysis, Complex Analysis, Functional Analysis, and Harmonic Analysis are "pure" math classes. But I disagree, since it is obvious that Number theory is Integral(basically a pun kinda) to cryptology, Topology has applications to chemistry, engineerung, and other areas. Real Analysis has been mentioned by Economists and Computer engineers as relevant. But if I could name 1 job for students who Love Analysis, it is Digital Signal processing. It is certwin that this wngineering job uses hella math. Not just Analysis, of course, but Stats, Linear Algebra, Discrete math, and others.
I have also seen peer-reviewed articles by Chemists commenting on how Abstract algebra is also relevant. I am of the opnion, that bon of the math classes typically at the undergraduate level are truly "theoretical/hypothetical". Every single undergrad class has applications as I have observed. It is, of course, obvious to yourself as a math educator, that math applies to music as well, so therefore Harmonic Analysis can have a confortable seat there as well. But even at the grad level, many math classes labeled "pure" have applications.


In short, the lines dividing "pure/theoretical/general mathematics and Applied are rather blurry. Ehat Math classes would you say truly have no known application and can be truthfully labeled as "pure?"