Assumptions

• Decimal addition and subtraction
• Decimal multiplication by 10

Proof

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