There are hordes of good books in all fields of mathematics. what you need is something you can learn from, not just the best and most glorious of these books.
books with so many problems and exercises with their suggestions and solutions are very attractive. but what you really need is a mature and deep understanding of the basics. after all, that’s all you need to tackle even these exercises with a surprising level of ease and fun.
You are reading: Best books on real analysis
analysis is among the most accessible fields in mathematics after high school, and a fair amount of knowledge in most other fields is required for beginners.
I understand the emphasis on solutions because we all deal with self-study, at least sometimes, and solutions/suggestions are crucial to doing an evaluation of your own work. if you’re really serious, you’ll soon discover that what you really need are suggestions, not solutions. It goes without saying that suggestions or solutions are supposed to be the last resort, when there seems to be no way out. even then, it’s better to take a hint only partially. and by the way: when problems are tackled, it is when there seems to be no way out that the real learning process occurs.
i encourage you to take a deep look at zakon’s trillia group funded and free books: mathematical analysis i, followed by another volume, but for some basics math basics might be a good place to start. in the third mentioned book, this was mentioned:
Class tests over several years led the author to these conclusions:
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1: The earlier such a course is taught, the more time is gained in follow-up courses, whether it be algebra, analysis, or geometry. The longer students are taught “loose analysis,” the harder it becomes to get used to rigorous proofs and formulations, and the harder it becomes for them to shake the misconception that math is just memorizing and manipulating a few formulas.
2 – when teaching the course to first-year students, it is recommended to start with sections 1-7 of chapter 2, then move on to chapter 3, leaving chapter 1 and sections 8-10 of chapter 2 for last . Students should be encouraged to pre-read the material to be taught below. (Freshmen should learn to read math by rereading what initially seems “fuzzy.”) then, the teacher can limit himself to a brief summary and spend most of his time solving as many problems (similar to the assigned ones) as possible. this is absolutely necessary.
3 – early and consistent use of logical quantifiers (even in text) is extremely helpful. quantifiers are there to stay in mathematics.
4 – motivations are necessary and good, as long as they are brief and do not use terms that are not yet clear to the students.
In the second book, this was mentioned:
multi-year class tests led us to the following conclusions:
1 – volume i can be (and was) taught even to sophomores, though they only gradually learn to read and make rigorous arguments. a sophomore often doesn’t even know how to start a demo. the main obstacle remains the ε, δ procedure. as a remedy, we provide most exercises with explicit hints, sometimes with almost complete solutions, leaving only small “whys” to answer.
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2 – motivations are good if they are brief and avoid terms that are not yet known. diagrams are good if they are simple and intuitive.
3 – flexibility is a must. the course must be adapted to the level of the class. the “featured” sections are better differentiated. (continuity is not affected).
4 – “colloquial” language fails here. we try to keep the exposition rigorous and increasingly concise, but readable.
5 – it is advisable to have students read each topic beforehand and prepare questions in advance, to be answered in the context of the next lesson.
6: Some topological ideas (such as compactness in terms of open covers) are difficult for students. trial and error led us to emphasize the sequential approach instead (chapter 4, §6). “covers” are covered in chapter 4, §7 (“highlights”).
7: For students unfamiliar with the elements of set theory, we recommend our Basic Math Concepts as supplemental reading. (At Windsor, this text was used for a semester-long freshman preparatory course.) The first two chapters and the first ten sections of Chapter 3 of this text are actually summaries of the author’s basic math concepts topics. , to which we also relegate topics such as the construction of the real number system, etc.
I didn’t take these points very seriously, until I started reading and working on it. it’s hard to find yourself completely stuck somewhere – it seems like everything has been packed for a person learning on their own. suggestions are provided whenever necessary. on many occasions there are questions like “…why?” that help to follow the text rigorously.
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