Quote by matt grime
Sigh. Take this part:
I'm sure the point has been repeatedly made in this thread that this is completely wrong.
You are, as is almost always the problem. Using your intuition about what happens at every stage after a finite number of decimal places to assert something about the infinitely long decimal expansion. Your intuition is wrong. Infinity is not 'a really big real number'. It is not a real number.
Parallel lines do not meet in the Euclidean plane. If I'm wrong (and I'm not), then feel free to write down the point of intersection: hint there is no such point as infinity on the Euclidean plane.
The place to use points at infinity is projective geometry, and there need not be just one point at infinity.

Matt, I said finite, not fixed. My point is that it is error to refer to a point at infinity in Euclidean space, for a point to exist in Euclidean space, it must be a finite distance from the orgin. The definititon of parallel lines is that every point on one line is a fixed distance from the other line. I was refering to the face that a line can always be positioned so that every point is a fixed distance from another line. This has nothing to do with points at infinity.
1 = .99999... and 89.999... = 90 but there is no point at infinity so what is the problem?