Use this difference equation to show that u8 = 208 256 . * 12. Any number [0, 1] has a decimal expansion

Use this difference equation to show that u8 = 208 256 .

* 12. Any number ω ∈ [0, 1] has a decimal expansion

ω = 0.x1x2 . . . , and we write fk (ω, n) for the proportion of times that the integer k appears in the first n digits in this expansion. We call ω a normal number if fk (ω, n) → 1 10 as n →∞

for k = 0, 1, 2, . . . , 9. On intuitive grounds we may expect that most numbers ω ∈ [0, 1] are normal numbers, and Borel proved that this is indeed true. It is quite another matter to exhibit specific normal numbers. Prove the number