The value of a decimal number is by definition the sum of its digits weighted by the appropriate power of 10. If the fractional part is not finite, we need to

The value of a decimal number is by definition the sum of its digits weighted by the appropriate power of 10. If the fractional part is not finite, we need to compute an infinite series n 0.999… = lim Σ 9/10^k = lim ( 1 - 1/10^n ) = 1 - lim 1/10^n = 1 - 0 = 1 ^ n→∞ k=1 ^ n→∞ ^ n→∞ ^ ^ | ^ | ^ | | | definition | induction | algebraic basic now we're | | limit theorem proof getting silly 0.9 = 1 - 0.1 0.99 = 1 - 0.01 … = … which is the case of a geometric series ∞ Σ a⋅r^k = a / (1 - r) k=0 with a = 9/10, r = 1/10. Another way to look at this is via the constructive definition of the reals as equivalence classes of Cauchy sequences of rational numbers: The sequences (1, 1, 1, …) and (0.9, 0.99, 0.999, …) are just two of an infinite number of Cauchy sequences which make up the real number 1.