Books, Video lectures, other resources to Teach Yourself Analysis

My favorite basic analysis sources that I have found useful:

Textbook: Analysis with an introduction to proof, by Lay. This is probably the friendliest text book I have seen on basic analysis.

Principles of mathematical Analysis by Rudin is also very good, but a little advanced especially if you are teaching yourself. However, if you decide to go with Rudin, make sure you check out these companion notes. Even you do not buy the textbook, you can still read these notes. This is my favorite reference for learning analysis.

Online textbook: http://www.mathcs.org/analysis/reals/ . Very useful, lots of examples.

Online lectures:

Introduction to Analysis by Francis Su(Harvey Mudd)

When comes to introductory analysis the first name that comes to my mind is Analysis I and II by Terence Tao. Dr. Tao has a divine talent to write and teach mathematics (you can visit his blog too: What's New). You have my word, these two volumes worth the cost. You can google "Introductory real analysis filetype:pdf" and choose one that fits in what you are looking for. But, they contends will be quite similar, so try to pick one that cover basic notions of logic and (naive) set theory and having a huge amount of exercises.

Here's a link to great lectures by Vaughan Jones, another Fields Medal winner (as is Terence Tao). They are beautiful, self-contained, and very accessible. Generously available for free.

http://sites.google.com/site/math104sp2011/lecture-notes

Rosenlicht's Introduction to Analysis is an inexpensive and elementary book (it costs $8).