Anomalous Cancellation on 2-Digit Numbers/Examples/19 over 95

Example of Anomalous Cancellation on 2-Digit Numbers
The fraction $\dfrac {19} {95}$ exhibits the phenomenon of anomalous cancellation:


 * $\dfrac {19} {95} = \dfrac 1 5$

as can be seen by deleting the $9$ from both numerator and denominator.

This is part of a longer pattern:
 * $\dfrac 1 5 = \dfrac {19} {95} = \dfrac {199} {995} = \dfrac {1999} {9995} = \cdots$

Proof
Formally written, we have to show that:
 * $\ds \paren {\paren {\sum_{i \mathop = 0}^{n - 1} 9 \times 10^i} + 10^n} \Big / \paren {5 + \paren {\sum_{i \mathop = 1}^n 9 \times 10^i} } = \frac 1 5$ for integers $n > 1$.

So: