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So my limited mathematics education has been especially ignorant of analysis. In this vein, I'd like to teach myself some of the introductory basics.

I'm intrigued by sources that might contain video lectures to complement readings and problems. Self-contained packages of all of the above are especially welcome!

Free is always better, but if quality comes at a price, feel free to suggest pocket lightening options...

And I'm certainly not ignorant of MIT's (awesome) opencourseware! Options are always nice though...

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up vote 11 down vote accepted

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: . Very useful, lots of examples.

Online lectures:

Introduction to Analysis by Francis Su(Harvey Mudd)

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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.

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+1.The original lecture notes to the course,which eventually evolved into Tao's very nice texts,are still available for free at Tao's blog.I bring that up because cost was a concern to the OP. – Mathemagician1234 Apr 28 '12 at 22:39

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.

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

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It's also incredibly well-written. Not very comprehensive obviously, but great at what it does cover. – gogurt Jul 1 '13 at 18:25

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