When I was a student I was exposed to mathematical analysis thru the book _Principles of mathematical analysis_ by Walter Rudin. I must say that I loved it and I then became a working mathematician in the field of analysis also in view of that book. However I know that many students simply hate Rudin's book because it has no image at all, almost no examples, and only hard problems. Therefore I usually suggest Charles Pugh's book on Real Analysis (published by Springer-Verlag), a more gentle introduction to real analysis. I also like Tom Apostol's book on mathematical analysis ( **not** calculus!), but it may be hard to find nowadays.