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Unformatted text preview: -x ) + Z k < 1. Inductively, we may choose n 1 < n 2 < ... such that if k = 1 , 2 , ... then k ( x n k-x ) + Z k < 1 / k . By deﬁnition of the quotient norm as an inﬁmum, for each k there exists z k ∈ Z such that k x n k-x + z k k < 1 / k . Now ( x n k ) k is Cauchy and ( x n k-x + z k ) k is null, so ( z k ) k is Cauchy; say z k → z . Thus ( x n k ) k converges to x-z and so its parent (Cauchy) sequence ( x n ) n also converges to x-z . Done. 1...
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This note was uploaded on 07/08/2011 for the course MHF 3202 taught by Professor Larson during the Spring '09 term at University of Florida.
- Spring '09