0.999.../Proof by multiplication by 10

Assumptions

 * Decimal addition and subtraction
 * Decimal multiplication by 10

Proof


\begin{align} x          &= 0.999\ldots \\ 10 x      &= 9.999\ldots \\ 10 x - x   &= 9.999\ldots - 0.999\ldots \\ 9 x        &= 9 \\ x          &= 1.000\ldots \\ \end{align} $$