I am an undergraduate. Other undergraduates sometimes ask me to tutor them in an introductory real analysis course that covers the equivalent of the first half-dozen chapters of Rudin's Principles of Mathematical Analysis (baby Rudin). We meet once a week for about an hour to discuss concepts and their problem sets.
In the beginning of the semester, I seem to spend about half the time on concepts (mainly repeating definitions and giving examples) and the other half giving hints on the homework problems. Later in the semester (and in the second semester continuation), I spend most of the time giving them considerable help with the homework. I don't have any trouble explaining concepts. However, I have a lot of difficultly with providing the appropriate amount of help with problem sets. The biggest problem I see with my students is a lack of problem solving ability. Part of it is a result of a poor grasp on the material and part of it is laziness. They don't seem to be able to think about a problem constructively for more than 5 or 10 minutes before giving up. Further, many of the problems are relatively easy once you see the essential idea or 'trick'. So either I show them the 'trick' or they just get the answer from another student. I usually end up giving too much assistance.
I have suggested they try easier exercises to build up their problem solving skills. Abbott's Understanding Analysis is a good source of these. However, they are unwilling to do much extra work on their own. Of course, this is ultimately their problem, but as a busy undergraduate myself, I am sympathetic.
Another common concern is preparing for exams. A 1-hour exam is a much different format than an untimed problem set and much less forgiving. My standard advice is to first to memorize all of the definitions, be able to give examples and non-examples (where relevant), and memorize the statements of all of the important theorems. After they have done this, I suggest solving easy exercises from their textbook and trying to prove simple propositions in their book without looking at the book's proof (e.g. a finite intersection of open sets in $\mathbb R^n$ is open).
How should I address these issues? Are there any other techniques I could use to make my tutoring more effective? I am particularly interested in ideas that are specific to real analysis.