In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. It was presented to the faculty at the University of Gottingen in 1854, but not published in a journal until 1868. For many functions and practical applications, the Riemann integral can be evaluated by the fundamental theorem of calculus or approximated by numerical integration. The Riemann integral is unsuitable for many theoretical purposes. Some of the technical deficiencies in Riemann integration can be remedied with the Riemann–Stieltjes integral, and most disappear with the Lebesgue integral. Historically, the concept of Integration came into existence as a means of evaluating areas under a curve, i.e., in compliance with a geometrical need. The first rigorous approach was therefore, quite naturally based on intuitive ideas of a sum and in effect as the limit of sum, now-a-days known as Riemann sum. But when the limitations of this approach were exposed through different situations, a rigorous arithmetic approach was contemplated by G.F.B. Riemann (1826–1866) with remarkable success. This approach was known as Riemann theory of integration which plays a fundamental role in analysis. Although it can be assumed that readers are familiar with the concepts of bounded and topology of real numbers, even then in view of the importance and utility of these topics for the study of Riemann integration, we give some fundamental definitions and proof of main theorem.