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I know this question was asked many times but I have some specific questions. I know the usual recommendations but I am afraid of going for Rudin because I've read many reviews that said it wasn't good for self studying. I remember in one review I read the guy said " As you go through the book you get excited about some cool theorems and results only to find that Rudin gives a proof that only does the job and leaves out much of the intuition for you to either find on your own or look for elsewhere".

I don't like this type of texts because when the proof is too directed it becomes unsatisfactory. By directed I mean that the result is already established and we're just trying to make it formal by looking for arguments that just verify the fact without , for example , mentioning how one would first consider these arguments and how they would come up while trying to prove the result.

That said , There are other suggestions such as Barry Simon's comprehensive course in analysis. This is a new text which isn't reviewed a lot. The description says it may be suitable for a graduate level course but others say it gives a good introduction to the prerequisites but I'm not sure.

Any other suggestions ?

Edit : thanks for your answers

cmk
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Km356
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    Robert Burn's "Numbers and Functions" is amazing. – dfnu Jul 26 '19 at 11:40
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    royden is the standard text for real analysis at the graduate level. – Jürgen Sukumaran Jul 26 '19 at 11:44
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    Tao, Zorich, Bartle (Elements), Bartle (Introduction), Pugh, Hubbard & Hubbard, Abbott, Apostol (Calculus), Apostol (Analysis), Spivak (Calculus), Spivak (Calculus on Manifolds), Munkres, Royden, Rudin, Kolmogorov/Fomin, Stein/Shakarchi... the list goes on. There are many, many choices and they are all heavily recommended in various threads. Use several sources if you find yourself having trouble with a particular section, or ask here. – graeme Jul 26 '19 at 12:26
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    I will add that the benefit of going through a well-worn text like Rudin's PMA is that there are generally bountiful solutions and hints to be found online, including Professor Bergman's nearly 100 pages of notes, explanations, and supplementary exercises to accompany Rudin. – graeme Jul 26 '19 at 14:19
  • https://math.stackexchange.com/questions/2725690/textbook-on-intro-to-real-analysis?r=SearchResults&s=3|98.7755, https://math.stackexchange.com/q/594640/53259, https://math.stackexchange.com/q/2096062/53259, –  Jul 26 '19 at 20:19

3 Answers3

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I would purchase a copy of Understanding Analysis by Abbott. There are a lot of pictures and the exercises are aimed at undergraduate students.

Axion004
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Try "Introduction to Real Analysis" by Donald R. Sherbert & Robert G. Bartle

The book is extremely great and an absolute beginner can read and understand it with immense pleasure, It starts with basic sets function and ends up to Riemann integrals and some glimpse of Topology. No doubt it will help you to make you strong in this field.

nmasanta
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    This also has a good follow-up/alternative, in Bartle's "The Elements of Real Analysis." While it goes through some concepts in one real variable, it's primarily geared towards analysis in $\mathbb{R}^n$. – cmk Jul 26 '19 at 13:42
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I like "Understanding Analysis" by Abbott, as suggested by @Axion004. Alternatively, you could look at Tao's "Analysis I" and "Analysis II." The series starts from the ground (natural numbers, set theory, real numbers) and, by the end of the last book, works up to the Lebesgue integral. The books feature clear examples and explanations. and many of the simple propositions are left as exercises, which you might like for additional practice.

cmk
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