This little guy craves the light of knowledge and wants to know why 0.999... = 1. He wants rigour, but he does accept proofs starting with any sort of premise.

Enlighten him.

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[–] Tomorrow_Farewell@hexbear.net 5 points 4 months ago* (last edited 4 months ago)

So, under the relevant construction of the space of real numbers, every real number is an equivalence class of Cauchy sequences of rational numbers with respect to the relation R outlined in my comment. In other words, under this definition, a real number is an equivalence class that includes all such sequences that for every pair of them the relation R holds (and R is, indeed, an equivalence relation - it is reflexive, symmetric, and transitive, - that is not hard to prove).

We prove that, for the sequences (1, 1, 1,...) and (0.9, 0.99, 0.999,...), the relation R holds, which means that they are both in the same equivalence class of those sequences.

The decimals '1' and '0.999...', under the relevant definition, refer to numbers that are equivalence classes that include the aforementioned sequences as their elements. However, as we have proven, the sequences both belong to the same equivalence class, meaning that the decimals '1' and '0.999...' refer to the same equivalence class of Cauchy sequences of rational numbers with respect to R, i.e. they refer to the same real number, i.e. 0.999... = 1.