devotion & mathematics

                       One of our main aims in considering metric spaces is to study convergent sequences  in a context more general than that of classical analysis. The fruits of this study are many, and among them is the added insight gained into ordinary convergence as it is used in analysis.                                         Convergent and Cauchy Sequences : Contents : 1. Convergent sequence and limit point. 2. The limit point of a convergent sequence is unique  3. Cauchy sequence 4. Every convergent sequence in a metric space is a cauchy sequence                                      5. Complete metric space    1 .convergent sequence :-        Let X be a metric space with the metri...

 HISTORICAL BACKGROUND OF METRIC SPACES:                                      Classical analysis can be described as that part of mathematics which begins with calculus and, in essentially the same spirit, develops similar subject matter much further in many directions. It is a great nation in the world of mathematics, with many  provinces,  a few of which are ordinary and partial differential equations, infinite series, and analytic functions of a complex variable. Each of these has experienced enormous growth over a long history, and each is rich enough in content to merit a lifetime of study.                                      In the course of its development, classical analysis became so complex and varied that even an expert could find his way around in it only with difficulty....