This book is prepared not only for the students of pure and applied Mathematics but also for the students of all discipline. Mathematics is an essential and mandatory subject for all the students from their early stages. This book is also helpful for them because elementary definitions and properties on real numbers have been discussed here. I have tried to make the exposition of the fundamental principles clearly as well as concise without going unnecessary details and at the same time an attempt has been made to make the treatment as much rigorous and up-to-date as is possible within the...