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0.999.../Proof by multiplication by 10
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0.999...
Assumptions
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Decimal addition and subtraction
Decimal multiplication by 10
Proof
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${\begin{aligned}x&=0.999\ldots \\10x&=9.999\ldots \\10x-x&=9.999\ldots -0.999\ldots \\9x&=9\\x&=1.000\ldots \\\end{aligned}}$