QQ,
Infinity, as I'm almost sure someone has addressed, is a concept, not a number. So 1/infinity is at best a concept.
Since infinity is larger than any finite concept of magnitude, it follows that 1/infinity, to such an extent the concept is rigorous, must be smaller than any finite concept of magnitude.
Further if your number system (like the hyperreals or surreal number system) admits one transfinite number, then it admits an infinite number of them, and infinity conceptional is larger than all of the admitted transfinite numbers. Therefore it follows that 1/infinity must be smaller than any positive infinitesimal.
Therefore, one is tempted to declare that 1/infinity = 0. But this does not promote the concept of infinity to the status of number. Nor does it admit 0 times infinity equals one.
The Riemann sphere conceptionally adds just one transfinite number, called $$\infty$$, to either the real numbers or the complex plane, but in doing so it it forces to strip infinity of the property of being larger than all real numbers, for it is also smaller than all real numbers and $$-\infty = \infty = \sqrt{-1} \infty$$.
http://en.wikipedia.org/wiki/Riemann_sphere#Arithmetic_operations
1/(any transfinite number) is a specific non-zero infinitesimal in systems that permit such objects. This is surely what your PhD authority in Pure Mathematics was trying to convey. [Citation of that purported authority requested, please.]
1/(the point on the Riemann sphere identified by the symbol $$\infty$$) = 0, but this does not mean $$0 \times \infty = 1$$ or even $$\infty - \infty = 0$$ so some violence to our concept of number has occurred.
But if we agree to work with real numbers and their completeness axiom, then it is required that the only possible "completion" of the unending sequence 0, 0.9, 0.99, 0.999, 0.9999, ... is 1. This claim doesn't rest on which form of the completeness axiom you choose, it's the same if you use Cauchy sequences and limits, supremum of the infinite set, topological continuity, etc.
I disagree with the claim that "0.999... = 1" is a no-brainer. You need practice in applying formal logic to know that if formal logic decides this, then deciding it once decides it for all time, and that one's choice of definition for number matters in important ways. This is what [post=3127165]Hamming meant when he wrote "In mathematics we do not appeal to authority, but rather you are responsible for what you believe."[/post] As someone with a physics background, I have a geometry background, which means my concept of number is closely related to my concept of length and I can conceive of lengths in the ratio $$1\, :\, 2$$, $$1\, :\, \frac{2}{5}$$, $$1\, :\, \frac{1}{3}$$, $$1\, :\, \sqrt{2}$$, and $$1\, :\, \pi$$. In geometry, you can't have two points next to each other with no room between them to add a third distinct point. So the concept of magnitude derived from geometry, the real numbers, cannot admit a difference between 0.999... and 1. They are two names for the same point on the real number line.
To calculate 0.999...
ab initio one needs a general mechanism for evaluating an infinite sum of ever smaller pieces. That mechanism, both in analysis and non-standard analysis, is the limiting value of the sequence of finite partial sums as the number of terms in the partial sums grows without limit. (Thus the concept of infinity is used without requiring that we describe infinity as a number.) And that limiting value in this particular case is 1.
If x is a number different than 1 and y is a number smaller than | 1 - x |, then at most only a finite number of the partial sums will be in the y-radius neighborhood of x. Thus a number other than 1 cannot be the limiting value of the sequence of partial sums.
