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.
this post was submitted on 02 Jul 2024
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chapotraphouse
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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.
"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:
"⇒" 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.