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Unformatted text preview: MATH 602, WINTER 2009 4. COMPLETENESS Cauchy sequences in metric spaces were defined in 1. Let x j , j = 1 , 2 , . . . , be a sequence of points in a metric space ( M, d). (i) If x j converges, then it has the Cauchy property. (ii) If x j is a Cauchy sequence, then it is bounded. (iii) If x j is a Cauchy sequence and has a convergent subsequence, then x j itself converges. In fact, for s chosen so that d( x j , x ) < / 2 whenever j s , the triangle inequality gives d( x j , x l ) d( x j , x ) + d( x, x l ) < whenever j, l s , proving (i). Similarly, for s chosen so that d( x j , x l ) < 1 whenever j, l s , and any z M , we have, for all j , d( z, x j ) max(d( z, x 1 ) , . . . , d( z, x s- 1 ) , d( z, x s ) + 1) , as d( z, x j ) d( z, x s ) + d( x s , x j ) < d( z, x s ) + 1 if j s , and (ii) follows. Finally, if s is selected so as to satisfy the condition d( x j , x l ) < / 2 whenever j, l s , and x is the limit of a convergent subsequence, that is, d( x l , x ) < / 2 for infinitely many l , then, fixing one of these infinitely many l , we have for each j s the inequality d( x j , x ) d( x j , x l ) + d( x l , x ) < , which establishes (iii). By (iii), every compact metric space is complete (for definitions, see 1). The Bolzano-Weierstrass theorem for Euclidean spaces ( 3), combined with (ii) and (iii), implies in turn that every Euclidean space is complete . Consequently, so is every finite- dimensional normed vector space , since, in a finite-dimensional space, any two norms are equivalent ( 3), while two equivalent metrics obviously have the same Cauchy sequences. We call a nonempty subset of a metric space complete if it is complete as a metric space in its own right (with the restriction of the original distance function). Every complete subset is closed (in view of (i) and uniqueness of the limit). Thus, according to the last paragraph, in a in a normed vector space ( V, k k ) , every finite-dimensional subspace is closed , and so if dim V < , the only dense subspace of V is V itself , while no finite-dimensional...
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