Good book for self study of a First Course in Real Analysis

When I was learning introductory real analysis, the text that I found the most helpful was Stephen Abbott's Understanding Analysis. It's written both very cleanly and concisely, giving it the advantage of being extremely readable, all without missing the formalities of analysis that are the focus at this level. While it's not as thorough as Rudin's Principles of Analysis or Bartle's Elements of Real Analysis, it is a great text for a first or second pass at really understanding single, real variable analysis.

If you're looking for a book for self study, you'll probably fly through this one. At that point, attempting a more complete treatment in the Rudin book would definitely be approachable (and in any case, Rudin's is a great reference to have around).

I like Terrence Tao's Analysis Volume I and II. By his simple way of explaining things, this book must be readable by yourself.

You can see here http://terrytao.wordpress.com/books/ all his books along with the two, I mentioned above.