Real Analysis

The density property of real numbers says that between two real numbers there is always an another real number. For example, 1.22 lies between the two real numbers 1.2 and 1.3. In this post, we will state and prove this density property of real numbers. Statement of Density Property of Real Numbers Between any two … Read more

The set of rational numbers is dense in the set of real numbers, that is, Q is dense in R. This is regarded as the density property of rational numbers. Before we proof this property, let us recall what are rational numbers. A number of the form p/q with q≠0 (p, q integers) is called … Read more

The Archimedean property of real numbers states that for positive real numbers x and y, there is an integer n>0 such that nx>y. This principle is named after the ancient Greek mathematician Archimedes. In this post, we will learn about the Archimedean property of real numbers along with its proof and applications. Statement of Archimedean … Read more

In this article, we will learn about the completeness property of real numbers with its definition, examples, and applications. Before starting it, let us note down a few important definitions. ℝ: The set of real numbers. Let S be a subset of ℝ. Note: The least upper bound of S (when exists) is known as … Read more

In this article, let us learn about the supremum and infimum of a set containing real numbers along with examples. We will also provide their properties with proofs. R stands for the set of real numbers. At first, we will learn the supremum of a set with examples. Definition of Supremum Let S be a … Read more

In this article, we will study the intermediate value theorem (also known as IVT) for continuous functions. As an application to this theorem, we will also learn the fixed point theorem for continuous functions. To establish the intermediate value theorem, we will require Bolzano’s theorem on continuity. This theorem is given below. Bolzano’s Theorem on … Read more