Concept guide
Complete metric space: proving every finite metric space complete by the least positive distance, contrasted with (0,1]
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.