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So I have a couple of weeks to study what I want before heading back to college. I am debating whether to study multivariable calculus or to study how to write proofs in general.

For context, I have already taken two proof based courses: A bit of real analysis and proof-based linear algebra. I did rather poorly in real analysis, and I attribute this to the fact that I did not have a lot of experience writing analysis proofs beforehand. Even now though I don't feel that I can write great real analysis proofs. In contrast, I did rather well in linear algebra, and felt I understood it decently well with a bit of preparation. However, I found that a non-insignificant number of times during linear algebra I wrote down a proof that I convinced myself was good, and there was some issue that made it a false proof.

I hope it's clear how it can be confusing as to what I should study. It seems that I did well in linear algebra by looking at examples of proofs, but neither before nor after this practice do I feel comfortable with real analysis, and there are also clear weak spots that I have. I'm not sure which way to approach learning proofs.

My own personal question is what is appropriate to study in order to get better at writing proofs, multivariable calculus proofs, real analysis proofs, or reading a book that generally aids with proofs? To make this question more general so that it is appropriate for this medium, I guess I would ask, what is better for learning proofs thoroughly after some exposure - seeing examples of proofs and trying your own, or reading a book for teaching proofs and trying your own? The book I've had in mind while writing this is Bridge to Abstract Mathematics by Morash.

I hope this is appropriate to ask here. I don't know what better place to get this advice from than math.stackexchange.com.

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    $\begingroup$ My opinion is that you can start with a real analysis book because others topics are more advanced and probably the proofs are not shown in the same level of detail. To understand how to prove something you need examples and practice, there are different general strategies to prove something depending of the topic you are studying. In this way you can try to complete the book Understanding analysis of Abbott. $\endgroup$
    – user173262
    Commented Mar 20, 2017 at 3:42
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    $\begingroup$ To be honest I wouldn't know where to start with books to read. But keep in mind that we've all made mistakes in our process of learning to do formal logic. it just takes practice. As for analysis in general I suggest reading a good analysis book that makes sense for you. $\endgroup$
    – Sentinel135
    Commented Mar 20, 2017 at 3:43
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    $\begingroup$ The best way to learn to write proofs is to try writing them then get feedback / criticism / suggestions from people who are good at proofs. Maybe there should be a stackexchange site specifically for that, analogous to the code review stackexchange site. Or maybe you can post your attempted proofs here on math.stackexchange with a question like, "Is this proof correct?" $\endgroup$
    – littleO
    Commented Mar 20, 2017 at 3:58
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    $\begingroup$ Terrence Tao's analysis book starts from very basic proofs using the Peano axioms and works its way up. He claims in the foreword that it really helped his students get a handle on proofs. I remember back in real analysis wondering what the heck when we were supposed to prove 1+1=2, but having gone back and looked at it I found it very useful. $\endgroup$
    – Arby
    Commented Mar 20, 2017 at 4:11
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    $\begingroup$ @KennyDuran you're on the right track. 1 is the lowest upper bound. You can prove it's existence by noting that $x<1$ for all $x \in A$. You then need to prove that it is the lowest one. So suppose it isn't... $\endgroup$
    – Sentinel135
    Commented Mar 20, 2017 at 4:47

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I think you should study multivariable calculus. Learning to write proofs is not something you can do without relevant statements to prove, or at least it's a quite hard thing to do.

I would also recommend that you diversify your math courses. Proofs in different sub disciplines look a bit different and will improve your repertoire. Also different teachers will have different styles which will help you finally get it.

Also depth is useful in order to learn proving things. The proofs that you do in single-variable calculus are often special cases (because the theorems are) of proofs that you do in multi-variable calculus and beyond. This means some repetition (with variations) that is also beneficial. This is also a good thing because if you really didn't get it the first time you will when you get it have proven something more general and not actually need to prove the special case from single-variable calculus.

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