0.999.../Decimal multiplication by a small number

Multiplication of infinite decimals is usually challenging because it involves a great deal of carrying. Fortunately, as in the cases of addition and subtraction, we are interested in identities that involve no carrying at all. Assumptions Definition from series Term-by-term operations on series Theorem Statement If there are two decimals A = a0.a1a2a3… and B = b0.b1b2b3… and an integer m such that for every index n, m × an = bn, then m × A = B. Proof We apply the definition of an infinite decimal as a series: Next we apply the fact that a scalar multiple of a series can be computed term-by-term: