posted on May, 7 2013 @ 01:55 PM
I didn't read through the whole thread, so perhaps someone actually got around to proving that
0.999... =
1 when working with
real numbers. This is not a matter of belief, nor does it actually have
to do with infinity as most people think of it. In actual mathematics, infinity is used as shorthand for states such as "for any arbitrary n>N some
condition is met." To comprehend why in the world 0.999... = 1, first make sure you know what 0.999... means. 0.999... does not mean 0.9 followed by
an arbitrarily large amount of 9's, nor does it mean 0.9 followed by an infinite number of 9's. To the first, 0.9 followed by any finite amount of
9's is certainly a number less than 1. To the second, see my comment above concerning infinity. When representing a number with decimal expansion,
use the definition of decimal expansion! Decimal expansion denotes a number that is the limit of the sequence (a, a b/10, a + b/10 + c/100,...). For
example, 2.25 is equal to the limit of (2, 2 + 2/10, 2 + 2/10 + 5/100, 2 + 2/10 + 5/100 + 0/1000, + 2 + 2/10 + 5/100 + 0/1,000 + 0/10,000). It is easy
to see that the limit of this sequence is 2 + 2/10 + 5/100 = 9/4.
But what is a limit? Again, go by the standard mathematical definition! A limit of the sequence (x_n) is a number L such that for any number E>0 there
exists an natural number N such that for any natural number K such that if K is greater to or equal than N, |L-(x_K)| < E. In English: The sequence
(x_n) is arbitrarily close (that is, less than any E) to L at some point in the sequence (that is, at the Nth term) and every point thereafter (any
Kth term after the Nth term). Furthermore, if a sequence has a limit, it is unique. There is no doubt that the sequence (0.9, 0.99, 0.999, 0.9999,...)
can be arbitrarily close to 1 simply by reaching some term in its sequence. It is impossible to show that there is a number greater to or equal than 1
that can possibly equal 0.999… as the sequence would either surpass the number and begin diverging from it or there would be a fixed gap between the
proposed number and the limit of the sequence (e.g., if you think 0.999… = 1.1, then you will never find a ϵ