0.999...

See also: 1

0.999... is a number that is equal to 0.9 + 0.09 + 0.009 + ...

Therefore, it is of the form $$\sum_{n=1}^{\infty}{9 \over {10^n}}=0.9+0.09+0.009+\ldots$$.

0.999... = 1
0.999... is known for making people confused with the fact that 0.999... = 1.

There are multiple proofs to make this true.

Confusion
The problem is some people think that 1 - 0.999... = 0.00...001. However, this is false as there is no last 0 for the last 1 to exist. So 1 - 0.999... = 0.000... = 0.

1/9 Proof
$$\begin{align}&\begin{align}{9\over9}&=0.999...\\{1\over9}&=0.111...\\{9\over9}&=1\end{align}\\&\operatorname{Q. E. D.}\end{align}$$

1/3 Proof
$$\begin{align}&\begin{align}{3\over3}&=0.999...\\{1\over3}&=0.333...\\{3\over3}&=1\end{align}\\&\operatorname{Q. E. D.}\end{align}$$