<
0.999...
Assumptions
Decimal addition and subtraction
Decimal multiplication by 10
Proof
x
=
0.999
…
10
x
=
9.999
…
10
x
−
x
=
9.999
…
−
0.999
…
9
x
=
9
x
=
1
{\displaystyle {\begin{aligned}x&=0.999\ldots \\10x&=9.999\ldots \\10x-x&=9.999\ldots -0.999\ldots \\9x&=9\\x&=1\end{aligned}}}
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