This edition is a modification for my first edition of ” A Simpler Approach to real Analysis” that I have used as a text book for my course in Math-3060 of Real Analysis on Spring 2011 at North Park University. The book is designed for students who have completed the ordinary course in elementary calculus, and it covers a portion of the material that every graduate student in mathematics must know. I hope that this book can enable the student to learn enough examples, theorems, and methods in analysis.