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Are there any analysis textbooks with concrete examples and problem sets?

I've studied mathematical analysis and real analysis with Rudin.

There was not much trouble for me in understanding what was written inside those textbooks. But recently I've found I have a problem while preparing for my GRE maths test. I know all the definitions and theorems in the textbooks but I'm not really able to apply them to real questions.

So for instance I know what a limit is and what traits it has, but if you give me a complicated function and ask me to find the limit I'm not really able to do so. I think maybe this is because abstract theory and application are slightly different and even if I know the theory I need some practice in application to solve problems for my GRE test.

And I don't think Rudin will be the one who can help. So do any of you know good textbooks that will help me prepare in learning the 'practical' maths?

Chloe RL
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  • Related: http://math.stackexchange.com/questions/97261/need-help-any-good-textbook-in-undergrad-multi-variable-analysis-calculus?rq=1 and here http://math.stackexchange.com/questions/345481/real-analysis-textbok-that-develops-the-subject-in-a-self-motivated-coherent-fa?rq=1 and here http://math.stackexchange.com/questions/373401/self-study-real-analysis-tao-or-rudin?rq=1 and here http://math.stackexchange.com/questions/91170/books-to-study-for-math-gre-self-study-have-some-time?rq=1 and so on. – Enrico M. Aug 21 '16 at 16:49
  • It is surprising to me if you are talking about solving $\frac{0}{0}$ and $\frac{\infty}{\infty}$ type problems, part of high school lore while Rudin is university lore. The essential idea is that the numerator must balance the denominator in a limiting sense as $\frac{0}{0}$ is uncomputable. So you simplify the expressions in numerator & denominator & cancel out whatever is making the expressions $0$. – Ricky Aug 21 '16 at 16:56
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    @Ricky: can you quote the sentence where the OP refers to "solving $\frac{0}{0}$ and $\frac{\infty}{\infty}$ type problems"? – symplectomorphic Aug 21 '16 at 18:03
  • @symplectomorphic: I don't know what is so tough then after mastering Rudin. – Ricky Aug 21 '16 at 18:20
  • @symplectomorphic: Also, the definition of derivative i.e. $\frac{dy}{dx} = \lim_{\delta x \rightarrow 0}\frac{\delta y}{\delta x}$ is nothing but a $\frac{0}{0}$ problem. – Ricky Aug 21 '16 at 18:30
  • @Ricky I'm not saying I have trouble with calculating limits only, that was just a simplified example to illustrate the kind of problem I have:) – Chloe RL Aug 22 '16 at 00:37
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    One of the more common lines that math teachers hear students say is: "I really understand the theory, I just can't do the problems." And I can tell you that in 99.99% of those cases, the student does not really understand the theory. In fact, often the reality is that they understand it so poorly, they don't even realize that they don't understand it. – Matt Brenneman Sep 17 '17 at 13:34
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    Does this answer your question? Exercise book for Elementary/Introduction to Real Analysis? –  May 04 '23 at 00:38

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A book I'm going through that has helped me a lot is "How to Think about Analysis" by Lara Alcock (Oxford University Press).

jim
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