Proof showing that the nonterminating decimal .1348888… represents a rational number. Please verify.

Question

My proof goes like this: Let $x=0.134888⋯$

$1000x=134.888⋯$

$1000x=134+0.888⋯$

$10000x=10(134+0.888⋯)$

$10000x=1340+8.888⋯)$

$10000x−1000x=1340+8.888⋯−(134+0.888⋯)$

$9000x=1340−134+8.888⋯−0.888⋯$

$9000x=1214$

$$x=\dfrac{1214}{9000}=\dfrac{607}{4500}$$

Then I concluded that this is a rational number. Any feedback is greatly appreciated. Thank you,

Answer 1

Note that ${1 \over 9} = 0.\bar{1}$, so $0.\bar{8} = {8 \over 9}$.

Then $0.134\bar{8} = {134 \over 1000} + {1 \over 1000} {8 \over 9}$.