PLEASE DO NOT USE THIS TEXTBOOK AS YOUR PRIMARY SOURCE TO STUDY REAL ANALYSIS IN ITS CURRENT STATE. IT IS UNFINISHED AND UNREVISED, AND MAY NOT BE ENTIRELY ACCURATE.
If you need a strong primary source, I would recommend the source referenced below: "Understanding Analysis" by Stephen Abbott. I consider it a more manageable book to start with.
THIS TEXT IS INTENDED TO BE READ IN "DARK MODE". Reading it in light mode shouldn't necessarily cause any problems, but the colored version may be more difficult to read, rather than less.
This textbook is my personal effort to make Real Analysis a more accessible course. It aims to take a pedagogy-first approach to mathematics: it concerns itself less with the most proper way of depicting mathematics, and instead leans into the intuitions and informal language that students are much more comfortable with.
The material is based on "Understanding Analysis" by Stephen Abbott, which, to the best of my knowledge, largely contains the same information as the traditional Real Analysis textbook, "Principle of Mathematical Analysis", by Walter Rudin. I chose this textbook as a basis because of its greater focus on accessibility: a feature I want to lean into with my own work.
As beautifully and concisely as Rudin's book is written, it is often too dense or borderline obtuse for a student fresh to Analysis, and especially one who does not possess prior 'mathematical maturity'.
Given that this course is often used to foster mathematical maturity at the same time, this seems like a student-unfriendly choice, and anecdotally, it is: many students give up on math at this point in their journey, despite having strong logical skills. They are often discouraged by jargon, heavily abstract ways of thinking, and material that is designed from the perspective of an experienced professor, not a fledgling math student.
Abbott's text, despite its strengths, is not consciously focused so much on this accessibility, either: its introduction suggests that it's more focused on the question of which topics that should be introduced to the student early, and things like that.
So, this textbook will face square the problem of introducing real analysis, in a way that feels more natural and concrete to new students.
Often in this text, traditionally concise proofs are "unpacked": at each step, we want to answer the question, "how could I have come up with that step on my own?" We suggest lines of inspiration, or the sort of problem-solving someone might use to develop their proof.
Students often come away from math feeling as if the words before them came from some genius they could never touch, and not just a mathematician using what they know in a clever way. This text hopes to pull back the curtain, and help a student figure out how they might write their next proof, and what sort of pitfalls might exist as they work. This way, they can see the inner workings of a proof, and not just the finished product.
Additionally, this text experiments with extensive coloring and highlighting of important or related words in a sentence: many different colors are used for different topics and ideas, in an informally and not fully consistent way. Coloring text like this is meant to make the textbook more easily legible, and to break up the walls of black and white. This is why dark mode is recommended. However, if these visual effects make you uncomfortable, or seem overwhelming, an uncolored version is provided.
This text is an ongoing project: at the time of writing this (11/28/2020), there is only 1 chapter of 8 written out. The author will continue to steadily advance through the text, hopefully over the next several months. Suggestions, corrections, or comments are welcome. Thank you for taking the time to read, and look into this little experiment.