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.
If A = 0.a1a2a3… then 10 × A = a1.a2a3a4…
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 Shifting a series|shift the series:
But a0 = 0 by assumption, so we can simplify:
which is the desired result.