0.999.../Proof by multiplication by 10

Assumptions

Decimal addition and subtraction
Decimal multiplication by 10

Proof

{\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}}

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