0.999.../Proof by multiplication by 10

Assumptions

• Decimal addition and subtraction Template:Stage
• Decimal multiplication by 10 Template:Stage

Proof

$\begin{array}{cc}x& =0.999\dots \\ 10x& =9.999\dots \\ 10x-x& =9.999\dots -0.999\dots \\ 9x& =9\\ x& =1.000\dots \\ \end{array}$