Both Pete-if you mind me calling you 'Pete',Dr.Clark,I apologize-and Jesse have given you terrific advice. I'd like to chime in and add my 2 cents on the matter.
Firstly,I also have Körner's book and I like it immensely.The book's frequently been attacked,strangely enough,for not being very original or creative in its choice of topics and organization. I do not understand this critique.It's a book for an intermediate undergraduate real analysis course-how creative could it be without defeating its purpose?!? It's a great supplement to little Rudin and asks a lot of the deep questions Rudin avoids. I also agree Gelbaum and Olmstead is a necessity on your desk for such a course and now that it's in Dover,there's no good reason not to have it.
Next,I'd like to recommend an alternative to Rudin. I know,the purists in the audience want to lynch me right now. Unfortunately,the sad fact of the matter is that most students today-especially in the United States-just aren't what they used to be in terms of discipline or background training. But my objection is deeper then that. Even if you're fortunate enough to have strong students,they really won't get as much out of the course using Rudin simply because it's so pristine and terse. They'll be able to do all the problems,understand the definitions and theorems. But I'll bet most of them walk out of there not understanding *why*it's necessary to build all this machinery into calculus.I saw a review of little Rudin at Amazon with some student praising the book as The New Testament of analysis and one of his comments gave me pause:" Don't let people steer you away from it with motivation. Motivation is for those too feeble-minded to handle real math. He's not selling you a car,Rudin-just do the book and the you will learn The True Way of Math." Or something to that effect. I think this attitude shows exactly what's wrong with using the book even with strong students.
The book I'm recommending was written-ironically-by Charles Chapman Pugh,who taught the honors analysis course at Berkeley for over 30 years. It's called Real Mathematical Analysis and I affectionately refer to it as Rudin Done Right. The book is pitched at the same level as Rudin, with exercises just as difficult if not more so.
So what's different about it?
Firstly,Pugh writes in a beautiful,conversational yet very concise style that's immensely personable,contains many references to the mathematical literature and yet is amazingly curt and direct. There's very little chatter.So how is Pugh able to be concise like Rudin and still be very informative and illuminating? Pugh has a remarkable gift as a textbook author:He seems to know instinctively exactly how many words it takes to explain something,not one word more, not one word less. For example,when defining the Cantor set,he refers to it as "Cantor dust". This phrase is extremely insightful in picturing the Cantor set for the first time.This intentional parsimony,where he carefully chooses,weighs and measures every word and phase,is one way he does it. The other is the book is full of pictures-there's literally a picture on every other page. But what's amazing here is that none of the pictures are throwaways or space fillers-every single picture is very specifically designed to make a point. For example,when proving that in a general metric space,every open ball is itself an open set,he presents it alongside a picture of an open disk in the plane and how a smaller open ball can be constructed anywhere inside an open ball. The book is loaded with beautiful pictures like that and if Rudin had done half of this in his presentation,the book would have been so much more approachable. It also covers certain topics vastly better then Rudin-his chapters on function spaces and their applications to differential equations,theoretical multivariable calculus and the Lebesgue are infinitely better then Rudin's.
Run right out and order this book for your students.You'll thank me later.Trust me.
Lastly,to the very good advice given to you by Pete and Jesse, I'd like to add one more thing. I strongly suggest you take a few lectures to build the real numbers from the integers for your students. Yeah,I know-a lot of analysts right now are screaming at me. But hear me out. I think one of the reasons students struggle with doing calculus rigorously for the first time is that they don't really understand the properties of the real numbers. Oh sure,they get the 10 axioms,they learn them-and then a lot of them still struggle with deriving inequalities. This,in my opinion,is really where the average student gets lost when doing analysis for the first time. They don't really understand the Cauchy-Schwarz inequality or why it's important in deriving the Triangle inequality,they don't really "get" why there's a rational number in between any 2 reals,why there's an integer N > 0 such that for any 2 reals x and y, Nx > y-you get the idea. After I took the time to construct the real numbers from scratch,I never had any trouble doing it. The big objection a lot of mathematicians have is that the derivation would eat up an entire semester by itself. Well,if you spell out and spoon feed every little detail of the construction, sure,that's true. But I don't think you have to do this to achieve the experience needed. Only certain steps need a lecturer to help the students through it-the rest is just spade work they can-and should-do if properly supplied with hints. Clearly,the jump from the rational numbers to the real numbers should be done carefully and in full detail.I think the earlier steps-building the natural numbers from the Peano axioms,etc.,can be done with far less detail if the key steps are done.Filling in these gaps can be the basis for wonderful problem sets for the students. Terence Tao's analysis course at UCLA famously was done that way.Tao found the 2-3 weeks or so work that was needed to do it more then paid off in the depth of understanding his students then had-the standard struggling with limits was almost completely avoided and the latter parts of the course not only went much more rapidly then in a standard class, there was much greater overall understanding by the students. I was absolutely floored reading about this in the preface to Tao's Analysis I and Analysis II books and I'm itching to try it myself when I teach real analysis the first time. I strongly suggest you try it and see how it goes.
(In fact,Pugh rather hastily sketches the development in the first chapter-giving a nicely detailed account of building R from Dedekind cuts of rationals where most texts prefer convergent Cauchy sequences of rationals.This is quite well done-although I don't agree with him calling Edmund Landau's Foundations of Analysis "classically boring".I think Landau's book should be required reading for all mathematics majors as they get ready for graduate school.)
Anyway-that's my very lengthy 2 cents on the matter. Good luck,let us know how it went!