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Most of the Students of mathematics find difficult to learn the essence of real analysis. This book is designed for under- graduate and post-graduate students of mathematics to understand the underlying principles and essence of real analysis. To achieve them solutions of a large number of examples are given on each topic. For example to understand the definition of limit of a sequence, solutions of various problems on limit of sequence are given in terms of ? and ?. Further they are verified with given value of ?. This helps to understand the concept of the limit of the sequence. This book consisting of five chapters. In the first chapter we discussed the real numbers, Dedekind-cut and mathematical induction. In the second chapter we discussed the interior point, limit point, open and closed sets. In the remaining chapters we discussed thoroughly on sequences, positive term series, alternating series and series of arbitrary terms