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I will be teaching a course which expects to have the following topics pulled from a now out-of-print book: Fundamental ideas of analysis, by Michael C. Reed, John Wiley&Sons,1998

As we can't use this book, I need something comparable.

The expected topics include (again these are presumably pulled from the book mentioned above, and I'm supposed to teach these):

  • Set of Natural Numbers
  • Set of Rational Numbers
  • Set of Real Numbers
  • Completeness Axiom

  • Sequences

  • Limit Theorems for Sequences
  • Monotone Sequences and Cauchy Sequences
  • Series

  • Alternating Series and Integral Tests

  • Continuity
  • Properties of Continuous Functions
  • Sequences and Series Functions

  • Uniform Convergence

  • Differentiation and Integration of Power Series
  • Mean Value Theorem

  • L’Hopital Rule

  • Taylor Theorem
  • Part one of Integration
  • Part two of Integration

So can someone tell me the most gentle undergraduate analysis book which would deal with these (as gently as possible)?

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Take a look at Understanding Analysis (2nd ed.) by Stephen Abbott. The book is pretty well-known these days for a very clear exposition of most of the basics of real analysis. I think most of the topics you mention are covered, if not all of them. In common with Bartle it gives a kind-of 'basics of topology in $\mathbb R$' in one of the early chapters to make some of the later proofs a little smoother. A nice thing about it is that every chapter starts with motivational example/s to demonstrate to the student why the material is worth studying. The only drawback is that no solutions manual exists to my knowledge... but if you're using it for a class this may be a good thing!

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I adore Spivak's Calculus. I think it covers every topic you mentioned in a completely rigorous fashion, and motivates the definitions (for example, he builds up to the definition of the limit until it seems like the only reasonable thing to do). It's also full of interesting problems.

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Introduction to Real Analysis 4e Bartle. It is the perfect transition from calculus to analysis; I was able to self-study from it. It is gentle as you have required and covers every topic you have listed and more such as basic prelimiaries, a generalization of the Riemann Integral, and a basic introduction to topology.

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I would still clear of Rudin, that's really more of a reference for people with a higher degree of mathematical maturity than I expect your students will have. The standard text used at my university was "An Introduction to Analysis", by Wade. It does use the axiomatic approach to completeness and the algebraic properties of the field of real numbers, instead of constructing them, which you've noted as a preference. It's also very good at providing counterexamples to demonstrate that the assumptions on theorems can't be relaxed. It is not as gentle as some others, but it is significantly more complete than others I've come across in terms of the material it contains. It should be within the grasp of someone who's already taken a course involving theorem proving (e.g. discrete math).

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Maybe a classic would be useful? Principles of mathematical analysis by W. Rudin is standard for those topics. I also like A Concrete Approach to Classical Analysis by Marian Muresan.

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    $\begingroup$ Rudin is not a gentle introduction. $\endgroup$ – zhw. Apr 22 '20 at 20:50
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I recommend Elementary Real Analysis by Thompson, Bruckner, and Bruckner. It’s free at http://classicalrealanalysis.info/com/ I believe the book is open source. The book doesn’t particularly cover the natural numbers or rational numbers but the rest is in book and the problems have a range of difficulties.

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