Reddit mentions of Analysis With An Introduction to Proof, 5th Edition

Pick up a book on proofs, and do as many exercises as you can. It's a big leap for a lot of students, going from plug and chug high school math to thinking creatively to prove results.

Also, learn some basic set theory (inclusion, union, intersection, cartesian product, etc).

Lay's Real Analysis book is very good for this purpose. It starts with basic logic and proofs, works in most of the set theory you need (though a bit light on Zorn's lemma), and then heads into the real analysis at a gentle pace. It also covers some basic topology along the way.

Not really, sorry. The only analysis textbook I'm familiar with is this one I used for my class, so I don't know how it compares to any others.

A book on Real Analysis or Discrete Mathematics would be good. You’ll be able to practice proving things with familiar topics like calculus and real numbers.

Very nice discrete book
Good intro to analysis of the real number line

From the preface of the first book
> Until this point in your education, mathematics has probably been presented as a primarily computational discipline. You have learned to solve equations, compute derivatives and integrals, multiply matrices and find determinants; and you have seen how these things can answer practical questions about the real world. In this setting, your primary goal in using mathematics has been to compute answers.
But there is another side of mathematics that is more theoretical than computational. Here the primary goal is to understand mathematical structures, to prove mathematical statements, and even to invent or discover new mathematical theorems and theories. The mathematical techniques and procedures that you have learned and used up until now are founded on this theoretical side of mathematics. For example, in computing the area under a curve, you use the fundamental theorem of calculus. It is because this theorem is true that your answer is correct. However, in learning calculus you were probably far more concerned with how that theorem could be applied than in understanding why it is true. But how do we know it is true? How can we convince ourselves or others of its validity? Questions of this nature belong to the theoretical realm of mathematics. This book is an introduction to that realm.

Any mathematical subject can be learned either as something applied, or something pure. That being said, if you're interested in pure math the main thing is learning proofs. That's the foundation of all higher level mathematics. If you can't fluently read (and eventually write) proofs, you aren't learning mathematics.

So, I'd suggest starting there. There are many books that are useful. In college, I used Lay's Analysis With an Introduction to Proof in a course, and found it very useful. Another thing I seen thrown around is the Book of Proof (pdf link there). I've never used the book personally, but it might give you a good place to start.

Then it's just a matter of going through various subjects. Discrete math (combinatorics, graph theory) is extremely accessible, and a pretty popular topic to begin learning proof with. You could also learn some abstract algebra (starting with group theory) as a more typical "standard" subject learned by math undergraduates. But really, if you want to learn math for its own sake, just find some books online and see what sticks. You have that freedom.

Here is an actual blog post that conveys the width of the text box better. Here is a Tufte-inspired LaTeX package that is nice for writing papers and displaying side-notes; it is not necessary for now but will be useful later on. To use it, create a tex file and type the following:

\documentclass{article}
\usepackage{tufte-latex}

\begin{document}
blah blah blah
\end{document}

But don't worry about it too much; for now, just look at the Sample handout to get a sense for what good design looks like.

I mention AoPS because they have good problem-solving books and will deepen your understanding of the material, plus there is an emphasis on proof-writing when solving USA(J)MO and harder problems. Their community and resources tabs have many useful things, including a LaTeX tutorial.

Free intro to proofs books/course notes are a google search away and videos on youtube/etc too. You can also get a free library membership as a community member at a nearby university to check out books. Consider Aluffi's notes, Chartrand, Smith et al, etc.

You can also look into Analysis with intro to proof, a student-friendly approach to abstract algebra, an illustrated theory of numbers, visual group theory, and visual complex analysis to get some motivation. It is difficult to learn math on your own, but it is fulfilling once you get it. Read a proof, try to break it down into your own words, then connect it with what you already know.

Feel free to PM me v2 of your proof :)