0.999.../Proof by equivalence of Cauchy sequences

Assumptions

• Construction of the real numbers from Cauchy sequences
• Definition from Cauchy sequences
• The limit of a geometric sequence

Proof

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.