Here's your fucking 7, snow-white

[math]

\displaystyle

\begin{align*}

\frac{1}{7} = \left (\frac{142857}{1,000,000} + \frac{1}{7,000,000} \right )

&= 0.142857 + \frac{1}{7,000,000} \\

= 0.142857 + \left ( \frac{142857}{1,000,000,000,000} + \frac{1}{7,000,000,000,000} \right )

&= 0.142857142857 + \frac{1}{7,000,000,000,000}\\

= 0.142857142857 + \left ( \frac{142857}{1,000,000,000,000,000,000} + \frac{1}{7,000,000,000,000,000,000} \right )

&= 0.142857142857142857 + \frac{1}{7,000,000,000,000,000,000} \\

= 0.142857142857142857142857 +\left ( \frac{142857}{1,000,000,000,000,000,000,000,000} + \frac{1}{7,000,000,000,000,000,000,000,000} \right )

&= 0. \underset{n}{ \underbrace{142857142857142857142857}} + \frac{1}{7, \underset{n}{ \underbrace{000,000,000,000,000,000,000,000}}} \\

&\vdots

\end{align*}

\\ \displaystyle

\Rightarrow 0.\overline{142857} = \frac{1}{7}

[/math]