What I am trying to achieve here is to present the current official math position. Not that I agree with it—it’s too generous to infinities for my taste, but who am I to judge—but I still want to explain this official math position.
It is possible that I am somehow wrong doing that, but still, I do try.
What I am basically saying is that because there is no finite positive real epsilon, such that 1-0.9999… would be equal to that epsilon, therefore those two should be equal.
If they weren’t equal, there would be such a positive epsilon, which would be equal to their difference. But there isn’t. If you postulate one such an epsilon, a FINITE number of 9s already yields to a smaller difference—therefore contradicts your assumption.
This is the official math position as I understand it. I might be wrong about that, but I don’t think I am.
Qiauchu_Yuan is a mathematician and I’m quite sure he’s familiar with the “current official math position”. I don’t think your presentation of it is wrong, but I think it’s unnecessary in this particular discussion :-). When you say there are no real numbers between 0.999… and 1 and therefore the two are equal, you are not disagreeing with QY but with a hypothetical person he’s postulated, whose knowledge of mathematics is much less than either yours or QY’s.
I see your angle now. Perhaps his angle, too.
What I am trying to achieve here is to present the current official math position. Not that I agree with it—it’s too generous to infinities for my taste, but who am I to judge—but I still want to explain this official math position.
It is possible that I am somehow wrong doing that, but still, I do try.
What I am basically saying is that because there is no finite positive real epsilon, such that 1-0.9999… would be equal to that epsilon, therefore those two should be equal.
If they weren’t equal, there would be such a positive epsilon, which would be equal to their difference. But there isn’t. If you postulate one such an epsilon, a FINITE number of 9s already yields to a smaller difference—therefore contradicts your assumption.
This is the official math position as I understand it. I might be wrong about that, but I don’t think I am.
Qiauchu_Yuan is a mathematician and I’m quite sure he’s familiar with the “current official math position”. I don’t think your presentation of it is wrong, but I think it’s unnecessary in this particular discussion :-). When you say there are no real numbers between 0.999… and 1 and therefore the two are equal, you are not disagreeing with QY but with a hypothetical person he’s postulated, whose knowledge of mathematics is much less than either yours or QY’s.