Most gentle introduction to undergraduate analysis 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)?

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!
