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I am taking a real analysis course next year and I want to start slowly preparing for that class now, so I hope you can help me.

The class is quite challenging and the fail rate is relatively high. The class uses Rudin's Principles of Mathematical Analysis.

To prepare for the class, I decided to go through Abbott's Understanding Analysis, Book of Proof by Hammack and work on inequalities. I read that it is important to be good with proving inequalities so can you give me advice on how I can do that? Are there books that can help me with that?

What else do you think I should work on, what books do you recommend? Is it necessary to go through a more rigorous calculus textbook? The courses I did in calculus used Thomas' Calculus, which I think is much less rigorous than something like Spivak's.

Would very much appreciate your advice.

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I would recommend practicing bounding things with inequalities, so when you encounter a problem that requires the use of the triangle inequality (it comes up fairly frequently), you can find a bound fast. Many of the definitions used in analysis deal with inequalities as well, such as convergence of a sequence, limit of a function, continuity, etc...

Also maybe if you bought Rudin's book before next year and went through the definitions (such as in Chapter 2, where there is a lot of terminology/definitions on point set topology), it would certainly help, as when your professor teaches it next year, you will already have been accquainted. If Rudin is too difficult to read, perhaps try an easier read, such as Steven Lay's Analysis With an Introduction to Proof. I found it easier to read than Rudin. The key differences in terms of topics covered are that Rudin covers most of the material in metric spaces, while Lay focuses on $\mathbb{R}$ in particular. Also, Rudin discusses the Riemann-Stieltjes integral, while Lay sticks with the Riemann Integral.

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Thank you very much for the answer. I will get Lay's book and try to get through it before the class starts next year. – please delete me Apr 19 '14 at 15:30
No problem. Good luck! – Sujaan Kunalan Apr 19 '14 at 15:49

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