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0.999.../Proof by equivalence of Cauchy sequences
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0.999...
Assumptions
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Construction of the real numbers from Cauchy sequences
Definition from Cauchy sequences
The limit of a geometric sequence
Proof
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In this formalism the task is to show that the sequence of rational numbers
$\left(1-0,1-{9 \over 10},1-{99 \over 100},\dots \right)=\left(1,{1 \over 10},{1 \over 100},\dots \right)$
has the limit 0.