## Introduction to Real Analysis

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

From basic to higher mathematics

In 20th century several challenging problems concerning real numbers have been solved. However, there remain countless simple looking problems which... 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 pq... 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: It is a set which has no member. This set may be specified by defining a property for... Click here to read more

is a natural number. Each natural number has a successor . Two natural numbers are equal if their successors are... Click here to read more

We shall be using capital letters and for the sets of numbers as specified below: , the set of natural... Click here to read more

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

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

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

Every subset of is a set of real numbers. We shall define upper and lower bounds for a non-empty set... Click here to read more

The field and order axioms for and various other concepts connected with these as given enable us to make algebraic... Click here to read more

Archimedean Property: Theorem: If , then for any there exist such that . Proof: When , the theorem is evident.... Click here to read more

Theorem: If and are two non-empty subsets of such that (i) , (ii) , then either has the greatest member... 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 Real Line: To every real number corresponds a unique point of a directed straight line... Click here to read more