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

${\displaystyle \left(1-0,1-{9 \over 10},1-{99 \over 100},\dots \right)=\left(1,{1 \over 10},{1 \over 100},\dots \right)}$

has the limit 0.