# Concept derivation guides

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Concept guide



## [Complete metric space: proving every finite metric space complete by the least positive distance, contrasted with (0,1]](/en/derivations/derivation-mko-complete-metric-space-01/?from=%2Fen%2Fderivations%2F%3Fq%3Dmko-complete-metric-space)

Prove that every finite metric space (X,d) with at least one point is complete (immediate for a single point; with at least two points, the least positive distance δ forces every Cauchy sequence to be ultimately constant). On X0={0,1,3} with d(x,y)=|x−y|, compute δ=1, give an ultimately constant certificate and an explicit non-Cauchy sequence; finally prove that the subspace Y=(0,1] is not complete: x_n=1/n is Cauchy in Y but has no limit in Y.

Complete metric space

derivation-mko-complete-metric-space-01





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