this post was submitted on 02 Jul 2024
57 points (100.0% liked)
chapotraphouse
13468 readers
808 users here now
Banned? DM Wmill to appeal.
No anti-nautilism posts. See: Eco-fascism Primer
Vaush posts go in the_dunk_tank
Dunk posts in general go in the_dunk_tank, not here
Don't post low-hanging fruit here after it gets removed from the_dunk_tank
founded 3 years ago
MODERATORS
you are viewing a single comment's thread
view the rest of the comments
view the rest of the comments
I suppose I will post one myself, as I do not expect anybody else to have that one in mind.
The decimals '0.999...' and '1' refer to the real numbers that are equivalence classes of Cauchy sequences of rational numbers (0.9, 0.99, 0.999,...) and (1, 1, 1,...) with respect to the relation R: (aRb) <=> (lim(a_n-b_n) as n->inf, where a_n and b_n are the nth elements of sequences a and b, respectively).
For a = (1, 1, 1,...) and b = (0.9, 0.99, 0.999,...) we have lim(a_n-b_n) as n->inf = lim(1-sum(9/10^k) for k from 1 to n) as n->inf = lim(1/10^n) as n->inf = 0. That means that (1, 1, 1,...)R(0.9, 0.99, 0.999,...), i.e. that these sequences belong to the same equivalence class of Cauchy sequences of rational numbers with respect to R. In other words, the decimals '0.999...' and '1' refer to the same real number. QED.
I like how compact this one is ;)