0.999.../Decimal multiplication by 10

Multiplying an infinite decimal by 10 is just as simple as multiplying an finite decimal by 10: every digit shifts one space to the left. Assumptions Definition from series Term-by-term operations on series Shifting a series Theorem Statement If A = 0.a1a2a3… then 10 × A = a1.a2a3a4… 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: Next we shift the series: But a0 = 0 by assumption, so we can simplify: which is the desired result.