Repeating decimals are irrational

Snipet from https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber-en.txt/download

...
Real Nunmber(ℝ)::= {x| x is represented by n-ary <fixed_point_number>, the
   digits may be infinitely long }

   Note: This definition implies that repeating decimals are irrational number.
         Let's list a common magic proof in the way as a brief explanation:
           (1) x= 0.999...
           (2) 10x= 9+x  // 10x= 9.999...
           (3) 9x=9    
           (4) x=1
         Ans: There is no axiom or theorem to prove (1) => (2).

   Note: If the steps of converting a number x to <fixed_point_number> is not
         finite, x is not a ratio of two integers, because the following
         statement is always true: ∀x,a∈ℚ, x-a∈ℚ

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