>>38READ THIS AND BE PERSUADED OF IT FOR IT IS THE GOSPEL TRUTH OF MATHEMATICS, AS FAR AS THE REAL NUMBERS ARE CONCERNED.
You wish to describe an infinitesimal number. Call it 1/infinity, or 0.000...0001, or whatever else. You wish to maintain that it is less than any other strictly positive real number, but more importantly, you wish to maintain that it is nonzero. call it x. Then the infinitesimal x is greater than 0. You might like to say that it is a real number, or something else. At any rate, if we show that it's merely equal to zero, we discard your claim. Suppose I write 1/(2*infinity). you would rightly claim that this is equal to your 1/infinity, while attempting to maintain that neither is equal to zero. But what is (1/(2*infinity))? It is just (1/2)(1/infinity). And there's only one real value I know of that's half of itself, and that's zero. This is not rigorous, but it is appropriate to your level of contemplation of the problem. It is also legitimate, since we obeyed regular arithmetic throughout.
THERE IS NO SUCH THING AS A REAL NUMBER WHICH IS LESS THAN EVERY POSITIVE REAL, YET GREATER THAN ZERO.
IN EXACTLY THE SAME SENSE THAT THERE IS NO INFINITELY LARGE POSITIVE REAL NUMBER, SO TOO IS THERE NO INFINITELY SMALL POSITIVE REAL NUMBER.
Let's beat this into your head some more, with respect to the original question.
x = 0.9999...
10x = 9.9999...
10x - x = 9.9999... - 0.9999...
9x = 9
x = 1.
While not RIGOROUS, this proof works, since it's not like the fallacious 1=2 proof where one divides by zero (zing). All that it does is to show that the thing we mean by .999... and by 1 have the same numerical value. 1/3 + 1/3 + 1/3 = .333... +.333... +.333... = .999... =1. Precisely 1. Nothing other than 1.
INFINITESIMALS ARE RIGOROUSLY BANISHED FROM THE REAL NUMBER SYSTEM.
Some history! Newton and Leibniz used infinitesimal quantities and came up with appropriate results. The rigorous difficulty arose in the fact that they divided by a quantity which was pretended to be nonzero, and later discarded like it was zero. It took the foundations of limits and functions much later to give calculus a firm(er) base. Have a mathworld link while we're at it:
http://mathworld.wolfram.com/Infinitesimal.htmlInfinitesimal: "A quantity which yields 0 after some limiting process"
'Infinitesimals are legitimate quantities in the non-standard analysis of Abraham Robinson.'-Wikipedia. Moreover, they were necessary imaginitive tools in the initial development of the calculus. The point of which you will now be convinced is this: .999... = 1 and your infinitesimal turns out to be zero when you try to do anything with it as a number in and of itself. If you want to comment on an non-archimedian field, you are free to read up on it and do so. But your object has no place in the real number system. .999... is in every possible sense, 1, and that's the truth. The object you are referring to has no reality in the only number system on which you are prepared to comment on it.
Here ends the sermon.