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I'm a self-learner in maths & starting with real analysis. I looked for many books and many suggestions here (like this and this and many more) & finally choose Tao's Analysis volume. It's beatifully written, giving a beautiful overview and connected one aspect with another.

But since my mathematical maturity is still not that developed, sometimes I had to struggle pretty hard in undestanding simple things (like "$\varepsilon$ follow the axiom of substitution for equality", etc.). Another challenge I'm facing is many of the propositions in this book are given as exercise. This might be good, but for newcommer like me, it's bit difficult.

So, I was looking for a companion book which might help me practice my understanding while reading this book. The ordering of this alternative book's chapter might not be the same as Tao's (not even the entire scope as I think when I start understanding analysis bit more I can start walking with Tao on my own), but very similar. I don't want to go with Rudin as I want to use it after finishing Tao as test of my understanding.

Thanks!

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  • $\begingroup$ You will have a hard time reading Rudin after Tao, I believe. Unless you read his second volume before Rudin. $\endgroup$
    – user370967
    Commented Jan 12, 2019 at 20:54
  • $\begingroup$ @Math_qed: I'm planning to complete both the volumes before starting Rudin. It's a beautiful book. Want to finish it quick but my ability stopping me. So asked this question if I get some help. Thanks for your suggestion. $\endgroup$
    – Beta
    Commented Jan 12, 2019 at 21:01

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Well, I really like the book of R.P. Burn: Numbers and Functions - Steps into Analysis.

Publisher: Cambridge University Press Online publication date: May 2018 Print publication year: 2015 Online ISBN: 9781316018392 https://doi.org/10.1017/CBO9781316018392

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  • $\begingroup$ This is really a very interesting book! Thanks for your answer!!! $\endgroup$
    – Beta
    Commented Jan 12, 2019 at 17:29
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The Way of Analysis by Robert Strichartz may suit your needs.

The author explains well what his intents are at the beginning of each chapter. Most proofs are often explained with not much left for the reader to figure out, and there's quite a bit of intuition accompanied with this explanations. There's even a preliminary chapter as well for the information concerning logical quantifieres, proofs, and sets.

https://www.amazon.com/Analysis-Revised-Jones-Bartlett-Mathematics/dp/0763714976/ref=sr_1_1?ie=UTF8&qid=1547313840&sr=8-1&keywords=strichartz+way+of+analysis

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  • $\begingroup$ Thanks Nectric! This is another wonderful book! I think I should follow both the suggestions given to this question. $\endgroup$
    – Beta
    Commented Jan 12, 2019 at 18:01
  • $\begingroup$ You're welcome, and have fun! $\endgroup$
    – Metric
    Commented Jan 12, 2019 at 18:22
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Spivak's Calculus is a classic texts for people who want a gentle introduction to real analysis.

The book contains many problem and there are completely solved exercise sheets online for all exercices. The author also included solutions/answers to selected problems in the back.

You can't go wrong with this text. Spivak learns to think you as a mathematician.

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    $\begingroup$ Thanks for the suggestion. I'll look into this book. $\endgroup$
    – Beta
    Commented Jan 12, 2019 at 21:15

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