TITLE: Teaching Introductory Real Analysis
QUESTION [40 upvotes]: I am currently helping teach an introduction to real analysis course at UC Berkeley. The textbook we are using in Rudin's "Principles of Mathematical Analysis" (aka baby rudin). 
I am trying to find ways to help the students understand the material better. My jobs include 

Writing solutions to the homework exercises
Finding other examples that can supplement Rudin, as Rudin sometimes doesn't present enough examples
Suggest Problems for the Midterm and Final Examinations

For (2), I have found Kenneth Ross's book (Elementary Analysis: The Theory of Calculus) very helpful. Does anyone suggest any other books, that do a good job in "holding your hand and walking through" with various concepts in analysis?
Also, this is my first time teaching, so I am wondering how a good solution set should look like. Should it just show the formal solutions, or should it also tell the students a little bit about how to approach the problem and build intution. Or is that too much writing that will distract and be unmotivating for the student to read through? 
Also, each week we assign about 10-14 problems, but only 3 problems are graded. My solutions are to the 3 graded problems. I believe there is some advantage to this system, but do you I am putting the students at a strong disadvantage if I do not write all of the solutions? Time is limited sometimes, but I plan on writing the solutions to some other problems other than the graded problems. So:
How do you decide which problems are worthy of writing solutions to (assuming time is limited and I don't have time to write solutions for all 14 problems)? 
Does anyone have good ideas to supplement Rudin(which I think is a great book, but my students may disagree) to the 40 student undergraduate class? This is a sort of a broad question. I'm wondering what others did in their previous experiences, if any, while teaching a class with that book?

REPLY [2 votes]: I really enjoyed Bergman's supplement to the exercises in Rudin's Principles. He also has a short set of notes to instructors and answers to questions from students. These helped me a lot in going through Rudin.
Also, I don't think they're at a disadvantage if you do not provide them with solutions. There are plenty to be found on the web if they want to check their work (hopefully after the homework has been graded).