The_root_of_all_evil said:
Maze1125 said:
0.999... = 1
That is fact, pure and simple.
Equivalent to, not equal to. If you've any proof of your PoV, I'd be happy to hear it - but simply stating it as a fact doesn't make it so.
I gave you an entire thread full of proofs.
But hey, seen as you didn't bother to read my post last time. Here [http://www.escapistmagazine.com/forums/read/18.85789-Interesting-fact-0-999-1] it is again.
I'll even copy the most rigorous out for you:
An infinite decimal is defined to be:
lim(as n->infinity)sum(from k=1 to n) (a_k * 1/10^k)
where a_k is the kth digit of the decimal.
Therefore, 0.999... is defined to be:
lim(as n->infinity)sum(from k=1 to n) (9 * 1/10^k)
So all we need to do is show that that is equal to one.
Which is true iff for all e>0 there exists an N such that for all n>N |1 - sum(from k=1 to n) (9 * 1/10^k)| < e
Now sum(from k=1 to n) (9 * 1/10^k) is a finite sum, and so we can calculate that
|1 - sum(from k=1 to n) (9 * 1/10^k)| = |1/10^n|
So we need to show that for all e>0 there exists an N such that for all n>N |1/10^n| =1 then |1/10^n| e>0, then let N = 1/e and then |1/10^n| N
Hence the claim that, for all e>0 there exists an N such that for all n>N |1 - sum(from k=1 to n) (9 * 1/10^k)| < e, is true.
So, by the definition of a limit, lim(as n->infinity)sum(from k=1 to n) (9 * 1/10^k) = 1
Therefore, by the definition of infinite decimals, 0.999... = 1
QED
