The statement of the title, is, in fact, meaningless, because it tacitly assumes that we can add-up "infinitely" many numbers, and good old Zenon already told us that this is absurd.
The true statement is that the sequence, a(n), defined by the recurrence
a(n)=a(n-1)+9/10n a(0)=0 ,
has the finitistic property that there exists an algorithm that inputs a (symbolic!) positive rational number ε and outputs a (symbolic!) positive integer N=N(ε) such that
|a(n)-1|<ε for (symbolic!) n>N .
Note that nowhere did I use the quantifier "for every", that is completely meaningless if it is applied to an "infinite" set. There are no infinite sets! Everything can be reduced to manipulations with a (finite!) set of symbols.
[math] \displaystyle
\frac{1}{3} = 0.\overline{3}= 0.1_3
[/math]
>>8116747
BRO 1/3 IS NOT EQUAL TO .3... YOU'RE ASSUMING WHAT YOU'RE ARGUING IS TRUE IN YOUR ARGUMENT YOU STUPID FAGGOT INFINITE PROCESSES DON'T EXISTS BECAUSE NO ONE CAN DO AN INFINITE PROCESS CAN YOU WRITE AN INFINITE NUMBER FOR ME NO YOU CAN'T SO YOUR PROOF IS RETARDED
>good old Zenon already told us that this is absurd
the fact that you can walk from your chair to the door is proof that you CAN evaluate the sum of an infinite series (halfway there, and half again, and again, etc.)
now stop posting, get up from your chair, go to the door, go outside, and never come back
>>8116797
[math]\displaystyle
0.1_3+0.1_3=0.2_3
[/math]
>>8116710
Pic related: the last thread on this topic
Don't get baited kids
>>8116710
From zero thread to this one fuck OP
>>8116710
> assumes that we can add-up "infinitely" many numbers
no one assumes that ,the 0.9999.. thing is about the convergence of the series
0.9 , 0.99 , 0.999 ,0.9999 ,0.99999.....
which converges to 1 as for any given number theres a spot in the series where it is arbitrarily close to 1 .
>>8116808
Not a stupid fuck finitist, but, your argument is invalid if spacetime is quantized as space would not be infinitely divisible.
>>8116710
>we can add-up "infinitely" many numbers
that is called a limit.
look at the infinite series:
[eqn]\lim_{n\to\infty}(\sum_{i=0}^{n}(0.9*10^{-i})[/eqn]
The proof that this series convergences against 1 is indeed trivial (there are also explicit formulas to calculate them).
Also notice how indeed every sum of the series represents only a finite amount of additions.
That means that finitely many additions can get you arbitrarily close to 1.
This more commonly knows as "equality".
>>8116797
>they don't exist
>therefore we can't do math on them
Someone needs to go back to middle school.
>>8116710
>0.9 = 1
>0.5 x 2 = 0.9
This is stupid.
You are wrong.
>>8117274
This.
"0.99999 != 1" memers should go back to the definition of Cauchy sequences.
>>8117709
"b-but if it goes on forever how will i get there"
They are like flat-earthers, stop responding to their garbage and hope that someday moderators will get tired.
x/9 = 0.x recurring
to get 0.9 recurring you'd have 9/9
i.e. 1
