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...

  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...

                    In the previous section we see the convergent and Cauchy sequences in a metric space with their properties. We see that a limit point is a point to which a sequence is convergent. Also every convergent sequence in a metric space is a Cauchy sequence and  the converse of the theorem i.e., every Cauchy sequence is not convergent.We illustrate it  with an example by considering (0,1] as metric space and a sequence {1/n} which is Cauchy but not convergent in (0,1] in detail. In the following there is an introduction to few points and few definitions. Now take a look upon them. Contents : 1. Limit vs limit point 2. Every where dense or dense set 3. Nowhere dense set 4. Boundary and boundary point of a set 1 . Limit vs limit point :-       Some times we often use the terms limit and limit points. Are they both have same meaning? This is a confusion to every reader.  On the real line...

                       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...