See Introduction to Calculus and Classical Analysis by Omar Hijab. The synopsis of the book on the publisher's website is as follows:
"This text is intended for an honors calculus course or for an introduction to analysis. Involving rigorous analysis, computational dexterity, and a breadth of applications, it is ideal for undergraduate majors. This third edition includes corrections as well as some additional material."
Some features of the text:
-The text is completely self-contained and starts with the real number axioms;
The integral is defined as the area under the graph, while the area is defined for every subset of the plane;
There is a heavy emphasis on computational problems, from the high-school quadratic formula to the formula for the derivative of the zeta function at zero;
There are applications from many parts of analysis, e.g., convexity, the Cantor set, continued fractions, the AGM, the theta and zeta functions, transcendental numbers, the Bessel and gamma functions, and many more;
Traditionally transcendentally presented material, such as infinite products, the Bernoulli series, and the zeta functional equation, is developed over the reals;
There are 385 problems with all the solutions at the back of the text.
See too Elementary Classical Analysis by Jerrold E. Marsden and Michael J. Hoffman. I believe the 'worked examples' you'll find there can help you in the lack of answers to the exercises proposed in your self-study journey.
[Reviewed by Allen Stenger, on 11/25/2012 ]
"This is an introductory text in real analysis, aimed at upper-division undergraduates. The coverage is similar to that in Rudin’s Principles of Mathematical Analysis and Apostol’s Mathematical Analysis. This book differs from these earlier books primarily in being more talkative: explanations are written out at greater length, there are more worked examples, and there is a much larger number of exercises at all levels of difficulty."