Mathematical analysis deals with concepts such as convergence, continuity, differentiation and integration. Such concepts must be based on an accurately defined number concept. In particular the real analysis is based on the real number system. So we begin our study with a discussion of the real number system.

We recall that a collection of objects viewed as a single entity will be called a set. The objects in the collection will be called elements or members of the sets. Sets are usually denoted by capital letters A, B. C, ... , and the elements by small case letters a, b, c,.... If a belongs to A, we write 'a ε A'.

The concept of point set topology grew out of the study of the real line and euclidean spaces. To be more precise, in this area of study we speak about sets of points on the real line, sets of points in the plane or sets of points in some higher - dimensional space.

In calculus, there are three basic theorems - Intermediate value theorem. Maximum value theorem and Uniform continuity theorem- about continuous functions and on these theorems the rest of the calculus depends. These theorems are used in a number of places. In fact, the intermediate value theorem is used for constructing inverse function.

Sequences form an important component of Mathematical Analysis. This concept was first studied rigorously by George Cantor and A. Cauchy. In fact they have shown that the sequences form most effective tools of Mathematics. We shall begin our study with some definitions and notations.

Suppose we are given two convergent sequences. It is natural to ask the questions related to the convergence of the sequences obtained by using rational operations between the two given sequences. Then the answer is affirmative. Similar questions can be asked when both the given sequences are divergent or when one is convergent and the other is divergent or when one is bounded and the other is con ......

In this unit we shall study the properties and applications of the monotonic sequences. We shall also introduce the upper and lower limits of sequences.

If a sequence {xn} converges to a limit x, then not only its terms ultimately become close to J; but also the terms become close to each other after certain stage. This observation leads to the definition of a Cauchy sequence. We shall prove in the following section that every convergent sequence is a Cauchy' sequence and also every Cauchy sequence is convergent.

The infinite series are merely special infinite sequences. Hence the discussion of the nature of infinite series can be made to depend on the nature of sequences. Like sequences, infinite series also form an important component of Mathematical Analysis and arise in man? situations.

In this unit we shall state and prove the comparison test, Cauchy's root test and D'Alembert's ratio test for the convergence of series. Some related results will also be discussed.

In this unit we shall discuss Cauchy-Maclaurin's integral test and Cauchy's condensation test. Further Kummer's test and its consequences will also be discussed.