0.999... equals 1

Example

The decimal expansion $0.999\dots$, with the digit $9$ at every place after the point, represents the real number $1$. By definition, $0.999\dots = \sum_{k=1}^{\infty} 9 \cdot 10^{-k}$. This is a geometric series with first term $9/10$ and ratio $1/10$, thus its sum is $\frac{9/10}{1 - 1/10} = 1$. Thus the number $1$ has two decimal expansions, $1.000\dots$ and $0.999\dots$.