0.999… always reminds me of my favorite definition of the real numbers:
“A real number is a quantity x that has a decimal expansion
x = n + 0.d₁d₂d₃…, (1)
where n is an integer, each dᵢ is a digit between 0 and 9, and the sequence of digits doesn't end with infinitely many 9s. The representation (1) means that
n + d₁/10 + d₂/100 + ⋯ + dₖ/10^k ≤ x < n + d₁/10 + d₂/100 + ⋯ + dₖ/10^k + 1/10^k
for all positive integers k.”[1]
Defining the reals in terms of binary expansion is left as an exercise for the reader[2].
[1] Knuth, The Art of Computer Programming, Volume 1, Third Edition, p. 21.
“A real number is a quantity x that has a decimal expansion
x = n + 0.d₁d₂d₃…, (1)
where n is an integer, each dᵢ is a digit between 0 and 9, and the sequence of digits doesn't end with infinitely many 9s. The representation (1) means that
n + d₁/10 + d₂/100 + ⋯ + dₖ/10^k ≤ x < n + d₁/10 + d₂/100 + ⋯ + dₖ/10^k + 1/10^k
for all positive integers k.”[1]
Defining the reals in terms of binary expansion is left as an exercise for the reader[2].
[1] Knuth, The Art of Computer Programming, Volume 1, Third Edition, p. 21.
[2] Ibid., p. 25, exercise 5.