Transcription of Complete Metric Spaces - Chula
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Complete Metric Spaces - Chula
Complete Metric SpacesDefinition (X, d) be a Metric space. A sequence (xn) inXis called aCauchy sequenceif for any >0, there is ann Nsuch thatd(xm, xn)< for anym n ,n n .Theorem convergent sequence in a Metric space is a Cauchy that (xn) is a sequence which converges tox. Let >0 begiven. Then there is anN Nsuch thatd(xn, x)< 2for alln N. Letm,n Nbe such thatm N,n N. Thend(xm, xn) d(xm, x) +d(xn, x)< 2+ 2= .Hence (xn) is a Cauchy converse of this theorem is not true. For example, letX= (0,1].Then (1n) is a Cauchy sequence which is not convergent Metric space (X, d) is said to becompleteif every Cauchysequence inXconverges (to a point inX).Theorem closed subset of a Complete Metric space is a Complete a closed subspace of a Complete Metric spaceX.)
Complete Metric Spaces Definition 1. Let (X,d) be a metric space. A sequence (x n) in X is called a Cauchy sequence if for any ε > 0, there is an n ε ∈ N such that d(x m,x
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