devotion & mathematics

  CONTENTS : Theorem  * : Cantor's  Intersection Theorem Theorem ** :  Let {x n } be a sequence in a metric space (X,d). For each positive integer n, write  E  n ={x  m /n ≥ m}. Then { x  n } is a Cauchy sequence in X if and only if lim diamE n =0. Theorem *** :  For any subset E of a metric space (X,d), diam E=diam Ē. The following are the proofs of the above statements : proofs : Theorem * : Cantor Intersection Theorem  Statement & Proof :     Theorem **  :  Let {x n } be a sequence in a metric space (X,d). For each positive integer n, write  E  n ={x  m /n ≥ m}. Then { x  n } is a  Cauchy sequence  in X if and only if lim diamE n =0. Proof :   Theorem ***  :  For any subset E of a metric space (X,d), diam E=diam Ē. Proof :                    people also ask :  1. What is a metric space?  2. What i...

                       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...