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  "body": "\n## Example\n\nThe [[decimal expansion|06XY4X9430]] $0.999\\dots$, with the digit $9$ at every place after the point, represents the real number $1$.\nBy definition, $0.999\\dots = \\sum_{k=1}^{\\infty} 9 \\cdot 10^{-k}$.\nThis is a geometric series with first term $9/10$ and ratio $1/10$, thus its sum is $\\frac{9/10}{1 - 1/10} = 1$.\nThus the number $1$ has two decimal expansions, $1.000\\dots$ and $0.999\\dots$.\n\n## Remarks\n\nAnother proof: the $n$th truncation is $1 - 10^{-n}$. The supremum of these numbers is $1$, by the [[Archimedean property|0A3GWZ86AY]].\nThe same happens for each number with a finite expansion. For example, $0.25 = 0.24999\\dots$.\n\n## Explanations\n\n### L1 human\n\nMany people feel that $0.999\\dots$ is a little less than $1$.\nBut the difference $1 - 0.999\\dots$ is smaller than $10^{-n}$ for every $n$.\nThe only nonnegative real number with this property is $0$.\nAnother view: $1/3 = 0.333\\dots$, and three times this gives $1 = 0.999\\dots$.\n",
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