... It's not that we "haven't found" a number strictly between .999... and 1. Rather, we can logically prove that there's no such number. That's a far stronger statement, with an entirely different meaning and philosophical significance. ...
while sleeping, my brain worked* a little more on this "Is there such an N?" and concluded NO (correctly as all who understand math will agree) but it also made the following puzzle about the extension (by induction, regression etc.) of the finite sum to an infinite number of terms:
rpenner in post 1603 first proves the validity of some intermediate steps I have replaced with ... in the following:
$$0.999... = \sum_{n=1}^{\infty} \, \frac{9}{10^n} .
... = 1$$ I'm sorry but can't get it to be what I want all on one line.
I don't do $$ so copied above from post 1603 and modified with the three dots, but now made a different mod, with a slight change. My change made the funny little "italic like v" sitting on top of the big summation sign. Not exactly what I expected, but ok so long as you understand the "script v" to be a finite number growing larger with n increasing.
I defined a finite sum < 1 , with funny little "script v" finite number of terms, the value of which is the function f(n):
f(n) =
$$0.999... = \sum_{n=1}^{\n} \, \frac{9}{10^n} < 1$$ and then define a slightly larger function of n, N(n), by:
N(n) = 1 - 0.5[1-f(n)] which is: 1 > N(n) > f(n) for all n.
Now the puzzle is that the very same recursive, or induction, etc. procedure that permit the claim that
$$0.999... = \sum_{n=1}^{\infty} \, \frac{9}{10^n} .... = 1$$
(Ignore three dots after the small n. With them in my poor "tex", at least it is all on one line.) also produces an N which is between 1 and 0.999...
I think this at least causes some doubt about applying that induction, recursive, (or what ever you wish to call it) procedure to extend the finite sum to an infinite number of ever smaller terms converging to the limit 1, without also producing that troublesome "inbetween N."
I.e. how does the extension procedure destroy the existence of this "in-between N", as n approaches infinity so that no N exist such that: 1>N>0.999...?
* BTW it is a fact that in some phases of sleep, the rate of energy use is as high as the peak rate during the awake state, so the idea that sleep is required to "rest the brain or save energy" is not much accepted. Nearly two decades ago, Crick (or perhaps Watson) of "twistied helix DNA" fame suggested we sleep to reorganize thoughts, memories, things learned during the day etc. when awake. In part this may have been related to fact too many "learing examples" when training up a neural network** computer can reduce its performance - over load or confuse it. It is primary, IMHO because of Crick's DNA fame that his POV about why we sleep is most accepted, but in truth we simply don't know why we sleep, especially as sleep is very common way down the animal kingdom. Sharks must forever swim as they don't have gill flaps to force water thru the gills like fish that can be stationary do. They too must sleep, but they do it "half a brain at a time."
I have my own idea as to why sleep is common: When sleeping the normal processing of sensory inputs is greatly suppressed, and motor commands even more so. I. e. In your dreams you can fight fierce battles with no related moves of your body. Likewise you can walk thru walls, get out of speeding cars, run thru fires, fly thru the air (that is quite common) etc. because evolution did not need to prevent such things. There are some restriction on your thoughts in sleep. For example you can not be in two different places in same dream at same time (and a few others which I forget) but in general you are with fewer restriction on your thought and logic. I think at least part of the reason we sleep it to "think-out side the box." On many occasions, like last night, I fall asleep, while thinking about some problem that has me "boxed in." When I awaken, I can often see a "hole in that box" I could not discover while awake. I. e. I think sleep evolved, at least in part to allow the mind to work on problems with less logical restrains or "known facts" limiting the process. That is what happened last night. I found what at least appears to be an "extension by induction etc." problem as the n approaches infinity for the convergent finite sum: 0.999999999999999999999999 (with a finite but large number of 9s) not explicitly written here.
SUMMARY: I hope someone more clever than me, will explain why N ceases to exist as 9.999... becomes 1. Until they can, there does seem to be at least some doubt that the induction (or regression etc.) process can be applied and infinite number of times.
** I hate that name for what should be called a "connection machine" or "connection computer." There are no neurons in any man-made (artificial) "neural network computer." In fact most are just simulated (emulated?) and trained up in modern digital computers. Some are then made into real electronic hardware, as for the special job they were train up on they are faster and much cheaper more compact and lighter weight. Most modern "clever torpedoes" use them for their intelligence for these reasons. I lost the battle to get the correct name used more than 4 decades ago.$$
