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

$(1-0,1-\frac{9}{10},1-\frac{99}{100},\dots )=(1,\frac{1}{10},\frac{1}{100},\dots )$

has the limit 0.