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

Theorem
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:
 * $\displaystyle \left({\left({\sum_{i \mathop = 0}^{n - 1} 9 \times 10^i}\right) + 10^n}\right) \Big / \left({5 + \left({\sum_{i \mathop = 1}^n 9 \times 10^i}\right)}\right) = \frac 1 5$ for integers $n > 1$.

So: