CANTOR'S INTERSECTION THEOREM #cantor #intersection #theorem #sequence #diameter #cauchy


  CONTENTS :

Theorem  * : Cantor's  Intersection Theorem

Theorem ** :  Let {xn} 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 diamEn=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 {xn} 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 diamEn=0.
Proof :
 

Theorem *** :  For any subset E of a metric space (X,d), diam E=diam Ē.

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