I don't see any reason why this "should" cover the reals. And it obviously doesn't -- it's trivial to prove that (1+sqrt(5))/2 isn't covered, for example, based on well understood bounds on how closely it can be approximated by rationals.
it's trivial to prove that (1+sqrt(5))/2 isn't covered, for example
It could be, depending on how the rationals are enumerated. I think the flaw is that while it's true that every real is arbitrarily close to an infinite number of rationals, the size of the segment centered of each of those rationals keeps falling, so "most" reals never get reached.
I actually agree with you, but we (you and I) have more formal background on this than most.
It's much more misleading if one draws the diagram - umbrellas centered on rationals, overlapping each other, it starts to look like they should all overlap everywhere. Clearly they don't, and you give an explicit example, but seeing how it fails can be difficult.
The more interesting thing is that we can put the "umbrellas" on the algebraic numbers, since they are also enumerable, and then it's harder to see exactly how it fails. However, start with one "umbrella" and look at the one centred on it's right edge, then the next, then the next, and you get a bounded, increasing sequence of algebraics. Again, we deduce the existence of the transcendentals.
But as you say, obviously it can't, but most people can't see the error in the argument.
The latest in the series, and probably the last about the real numbers. I've kinda lost track of the items and how they relate, but it all started in response to this:
