On 31 May 2007 01:02:12 -0700, firstname.lastname@example.org wrote:
>You have defined 0.999... to be a real number without jusitification. >I can make no such assumption. Each position within 0.999... can be >expressed as a real number, but the totality, the very infinite nature >of it, seems to render it a never ending relation more than a specific >explicit location on the real number line.
So what about sqrt(2)? or Pi?
Are those real numbers?
If so, note that they also have infinite decimals representations, thus are also "never ending". So why do you accept those infinite decimals as real but not .999... ?
For that matter, what about the fraction 1/3?
I'm sure you accept 1/3 as a real number.
Do you accept 1/3 = .333... ?
If so, note that .333... really means
3/10 + 3/10^2 + 3/10^3 + ...
which is the sum of an infinite series. Recall that the sum of an infinite series means, by definition, the limit of the partial sums, providing the limit exists.