Misconception??

The misconception is that $\color{#333333} 0.{\overline{9}}$ only approaches but does not equal $\color{#333333} 1$ . Now, what do we mean when we say that two numbers are different? There must be at least one more real number between them (in fact, there are infinite).

Since

$\displaystyle \color{#333333} 0.{\overline{9}} = 0.{\underbrace{999 \ldots \ldots \ldots \ldots}_{\text{Infinite number of 9s}}}$

there exists no number between $\color{#333333} 0.{\overline{9}}$ and $1$ , so it must be that they are the same. To show that $\color{#333333} 0.{\overline{9}} = 1$ , we have

$\color{#333333} \displaystyle \begin{aligned} x & = 0.999 \ldots \\ 10x & = 9.999 \ldots \\ 10x - x & = 9.999 \ldots \ - \ 0.999 \ldots \\ 9x & = 9 \\ x & = 1 \ \ \blacksquare \end{aligned}$