Integer which is Multiplied by 9 when moving Last Digit to First/Proof 2

Proof
From Integer which is Multiplied by Last Digit when moving Last Digit to First, $N$ is equal to the recurring part of the fraction:


 * $q = \dfrac {a_1} {10 a_1 - 1}$

where $a_1 = 9$.

Thus:
 * $q = \dfrac 9 {10 \times 9 - 1} = \dfrac 9 {89}$

Hence:

Decimal Expansion
The result follows.