0.999999... = 1?

What the fuck. Why is that true. They got different numbers in them.

You know, if this idiot won't shut up about not doing operations on an infinitely repeating decimal, I whipped up a proof using an infinite series.  Maybe that will shut him up.

0.999~ = SUM(n=1,inf,9/(10^n))  [The series is 0.9 + 0.09 + 0.009 + ...]

SUM(n=1,inf,9/(10^n)) = 9 * SUM(n=1,inf,1/(10^n))

9 * SUM(n=1,inf,1/(10^n)) = 9 * SUM(n=1,inf,(1/10)^n)

Known theorem: SUM(k=1,inf,r^k) = r/(1-r)

9 * SUM(n=1,inf,(1/10)^n) = 9 * (1/10)/(1-(1/10))

9 * (1/10)/(1-(1/10)) = 9 * (1/10)/(9/10)

9 * (1/10)/(9/10) = 9 * (1/9)

9 * (1/9) = 1

THEREFORE,

0.999~ = 1

QED

Sorry about the notation, but that's the best way to express it on a BBS.