I'm trying to self study real analysis because im starting my bachelor in math this year. Everyone says to go for "Understanding Analysis" but i strongly dislike it. the second edition doesnt have solutions so i have to use the first edition. And the proofs seem half baked. With steps that are unmotivated. So im looking for an introductory book that with the following criteria:
- free.
- has a (complete) solution manual.
- motivated proofs.
Prof. Robert Ash's Real Variables with Basic Metric Space Topology should satisfy your first two requirements, although its target audience seems to be physics or engineering students, not mathematics majors. Its prose is clear. Its coverage is broad (as an introductory text, it provides even an example of a continuous but nowhere differentiable function). Its proofs are rigorous and well-chosen (although the author seemed to opt for slick proofs rather than motivated proofs). The exercises have full solutions. The price is cheap (a free electronic version is downloadable from Prof. Ash's website and a low-priced print edition is available from Dover Publications). The text (in the print edition) is sharp. For non-mathematics majors, I think it is one of the best (if not the best) introductions to real analysis. For mathematics students, it is somewhat lacking in depth, but I think it is still a very good first text if one wants to get a taste for real analysis.