devotion & mathematics

COMPLETE METRIC SPACE :                                                        A metric space X with the metric d is called a complete metric space    if every cauchy sequence in x converges in X.     ∴ For Example : 1. consider the metric space (0,1) and {1/n} is a cauchy sequence in (0,1).                           We know that the sequence {1/n} converges to 0.                           since this limit point 0 does not lies in (0,1) , (0,1) is not a complete metric space.                                            Instead of (0,1)  if we consider  the metric space [0,1], th...

                       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...