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.

  • TheWurstman [he/him]
    ·
    3 days ago

    Doesn’t that just mean they’re both elements of a Cauchy sequence rather than equivalent? I suck at maths.

    • dat_math [they/them]
      ·
      2 days ago

      Not quite. The wording "equivalence classes of ... with respect to the relation R: aRb <==> lim( a_n - b_n) as n->inf" is key.

      https://en.wikipedia.org/wiki/Equivalence_class

      loosely, an equivalence relation is a relation between things in a set that behaves the way we want an equal sign to

      For an element in a set, a, the equivalence class of a is the set of all things in the larger set that are equivalent to a.

        • iie [they/them, he/him]
          ·
          2 days ago

          "Having the same age" is an equivalence relation between people.

          • It is reflexive: Bob is always the same age as himself

          • It is symmetric: if Bob is the same age as Sally, then Sally is the same age as Bob

          • It is transitive: If Bob is the same age as Sally, and Sally is the same age as Fred, then Bob is the same age as Fred.

          using symbols:

          Bob ~ Bob

Bob ~ Sally ⇒ Sally ~ Bob

Bob ~ Sally and Sally ~ Fred ⇒ Bob ~ Fred

          "⇒" means "the statement on the left implies the statement on the right." When people in this thread write =>, <=, and <=> they mean ⇒, ⇐, and ⇔

          An "equivalence class" is the set of all items that obey the equivalence relation with each other. So, "being 25 years old" is an equivalence class containing every person who is 25 years old. Those people might be different in every other way, but they are equivalent in that specific regard.

          In their proof earlier, @Tomorrow_Farewell@hexbear.net defined two equivalence classes. Instead of "people who are 25 years old," the classes were "infinite sequences that converge to 1" and "infinite sequences that converge to 0.999...." They showed that these are the same class.

    • Tomorrow_Farewell [any, they/them]
      ·
      3 days 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.