integers - how to proof that $1,9999.... in mathbb{Z}$?

I want to prove that $1,999\dots$ Is an element of $\mathbb{Z}$. Here is my try :

$x = 1,9999\dots \\ 10x = 19,9999\dots \\ 10x - x = 18 \\ 9x = 18 \\ x = 18/9 \\ x = 2$

So $x \in \mathbb{Z}$

I know something is wrong but where ?

Answer

$$x=1+9\sum_{i=1}^{+\infty}10^{-i}$$

$$=1+9\sum_{i=1}^{+\infty}(\frac {1}{10})^i$$

$$=1+9\frac {1}{10}\frac {1}{1-\frac {1}{10}}$$

$$=1+9\frac{1}{10}\frac {10}{9}=2$$