Are you saying that my logic is invalid?
Are you saying that it’s not true that we can know that an infinite product of (0) is (0) if we know that (0) raised to any number (whether finite or infinite) is equal to (0)?
How about an infinite product such as (1 \times 1 \times 1 \times \cdots)? How do you know the result of this product is (1)? Is it because we know that (1) times any quantity (whether finite or infinite) is (1)? Or is it because we know that (1) raised to any quantity (whether finite or infinite) is equal to (1)?
That’s probably because you’re deeply insecure and have a strong need to see flaws in people around you in order to feel good about yourself. And you’re looking for any kind of flaws, so as long they are flaws – big or small, significant or insignificant, etc.
Normal people don’t do that.
It’s of no help if what you’re doing is looking for a number that does not exist e.g. a finite number that is equal to the result of the infinite product (\frac{1}{10} \times \frac{1}{10} \times \frac{1}{10} \times \cdots). But if you’re merely trying to figure out whether such a number exists, then it’s quite a bit of help. It tells you that such a number does not exist.
The limit of an infinite product is not the same thing as its result. They are two different concepts.
It tells us that the result of the infinite product is smaller than every real number of the form (\frac{1}{10^n}) where (n \in N). Most importantly, it tells us that no matter how large (n) is, the result is always greater than (0).
Your argument is basically that there are no numbers greater than (0) but smaller than every number of the form (\frac{1}{10^n}) where (n \in N).
That’s one of our points of disagreement.
It does not merely “look like”. It will never ever get to zero for the simple reason that there is no number (n) greater than (0) that you can raise (\frac{1}{10}) to and get (0).
You can say that (0.\dot01) is approximately equal to (0), and that is true and noone disputes that, but that misses the point of this thread. We’re asking whether the two numbers are exactly equal not merely approximately equal.
You can say that (\frac{1}{0}) can be substituted with (\infty) for practical reasons (given that (0 \approx \frac{1}{\infty})) but you cannot say that (\frac{1}{0} = \infty) given that there is no number that you can multiply by (0) and get anything other than (0).
So from your point of view, the only conclusion that should make sense is that (0.\dot01) is a contradiction in terms, and thus, not equal to any quantity. By accepting such a conclusion, you’d have to agree that (0.\dot9 \neq 1). So at least one point of our disagreement (really, the main point of disagreement) would be resolved.
Still, one point of our disagreement would remain, and that would be your insistence that (0.\dot01) is a contradiction in terms based on the premise that there is no quantity that is greater than (0) but less than every number of the form (\frac{1}{10^n}, n \in N).