## Introduction to Real Analysis

In the 20th century, several challenging problems concerning real numbers have been solved. However, there remain countless simple looking problems… Click here to read more

From basic to higher mathematics

In the 20th century, several challenging problems concerning real numbers have been solved. However, there remain countless simple looking problems… Click here to read more

Mathematical Statement A meaningful composition of words which can be considered either true or false is called a mathematical statement… Click here to read more

Basic statements which do not seem to contradict themselves, to the best of human knowledge, are called axioms. It is… Click here to read more

To prove mathematical results, in general we use any of the following methods. (1) When statements of the form p$$… Click here to read more

A set is a collection of distinct and well-defined objects. In general, capital letters like A, B, G, S, T,… Click here to read more

Null Set This is a set which has no member. This set may be specified by defining a property for… Click here to read more

$$1$$ is a natural number. Each natural number $$n$$ has a successor $$n + 1$$. Two natural numbers are equal… Click here to read more

We shall be using capital letters $$\mathbb{N},\mathbb{Z},\mathbb{Q}$$ and $$\mathbb{R}$$ for the sets of numbers as specified below: $$\mathbb{N} = \left\{… Click here to read more

Axioms for Real Numbers The axioms for real numbers are classified under: (1) Extend Axiom (2) Field Axiom (3) Order… Click here to read more

Real numbers possess an ordering relation. This relation we denote by the symbol “$$ > $$” which is read as… Click here to read more

Sometimes it is useful to restrict our attention to non-negative real numbers only. For this purpose, we define a numerical… Click here to read more

Every subset of $$\mathbb{R}$$ is a set of real numbers. We shall define the upper and lower bounds for a… Click here to read more

The field and order axioms for $$\mathbb{R}$$ and various other concepts connected with these as given enable us to make… Click here to read more

Archimedean Property Theorem: If $$x > 0$$, then for any $$y \in \mathbb{R}$$ there exist $$n \in \mathbb{N}$$ such that… Click here to read more

Theorem: If $$A$$ and $$B$$ are two non-empty subsets of $$\mathbb{R}$$ such that (i) $$A \cup B = \mathbb{R}$$, (ii)… Click here to read more

Points on a straight line can be used to represent real numbers. This geometrical representation of real numbers as points… Click here to read more

Dedekind-Cantor Axiom of Continuity of a Real Line Evert real number corresponds to a unique point of a directed straight… Click here to read more

George Cantor (1845—1918), the creator of the set theory, made considerable contributions to the development of the theory of real… Click here to read more

$${k_2} \in \mathbb{R}$$ such that $${k_1} \leqslant {u_n} \leqslant {k_2}$$, $$\forall {\text{ }}n \in \mathbb{N}$$. Equivalently, $$u$$ is bounded if… Click here to read more

A number $$l$$ is said to be a limit point of a sequence $$u$$ if every neighborhood $${N_l}$$ of $$l$$… Click here to read more

The greatest and smallest limit points of a bounded sequence, as given by the preceding tutorial, are respectively called the… Click here to read more

Finite series generally do not involve any difficulty with respect to the validity of the application of algebraic operations as… Click here to read more

If $$\left\langle {{u_n}} \right\rangle $$ is given real valued sequence, then an expression of the form \[{u_1} + {u_2} +… Click here to read more

An infinite series $$\sum {u_n}$$ is said to converge, diverge or oscillate according to how its sequence of partial sums… Click here to read more

Consider the series \[ a + \frac{{{a_1}}}{{10}} + \frac{{{a_2}}}{{{{10}^2}}} + \cdots + \frac{{{a_n}}}{{{{10}^n}}} + \cdots ,\,\,\, – – – (*)\]… Click here to read more

We shall denote by $${S^ + }$$ the set of all such series whose terms are positive. Thus, $$\sum {u_n}$$… Click here to read more