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