I usually say "If and only if two numbers are different, then you can find a number between them". People often accept this axiom. Then, I offer them to find a number between 0.999... and 1.
We all wanna talk infinity because it sounds more exciting, but I think everybody gets "infinitely close to 1" pretty well intuitively. What they don't get is whether "infinitely close to 1" means "equal to 1". That could happen because these people are stupid.[note] But it could also happen if nobody has defined equality.
[note]for example even highly educated people maybe don't listen, which is functionally a lot like being stupid.
(also a somewhat joking answer)
Firstly, though, there are multiple different types of black hole, from the theoretical to the astrophysical. We must narrow your question down to have any hope of a good answer.
The simplest theoretical black hole, the Schwarzschild black hole, has one variable -- the central mass -- which must be positive.
If we set the central mass in a Schwarzschild spacetime to zero, then we have Minkowski spacetime: no curvature, no horizon, no black hole.
The Schwarzschild spacetime is completely empty except for the central mass, which is constant and located at an infinitesimally small point at all times. The Minkowski spacetime is completely empty everywhere and at all times.
The symmetries of Schwarzschild and Minkowski spacetime are different, and if one were to probe the spacetimes in question with a Synge curvature detector [1], we would quickly discover which we were probing if our probes happened to be placed close to the central mass, and eventually if they were placed far from the central mass.
If one placed the probes infinitely far from the central mass, it would take an infinitely long time to distinguish the presence of the central mass (which makes spacetime non-Minkowski); but these spacetimes are eternal anyway, so that's OK. So that's almost a "yes" to there being a theoretical black hole analogy between (1-) 0.999... and (1-) 1.
I would not call this a physical analogy since neither Minkowski spacetime nor Schwarzschild spacetime is at all physical. Nature is full of stress-energy (gas, dust, ...) any of which breaks the vacuum condition of these spacetimes, there seem to be a lot of astrophysical black holes at the centres of galaxies and individual/binary stars that have become black holes, and even a two-black-hole universe is markedly different than a Schwarzschild spacetime. Additionally, these astrophysical black holes are not eternal, unlike Schwarzschild. In particular, the stellar mass ones were once stars, and the galaxy-centre ones at least had less mass in the past. These last conditions alone are substantial deviations from Schwarzschild that are even more obviously not Minkowski (e.g. if you put probe finitely but sufficiently far away, you could see an image of the radiant precursor star rather than the black hole!).
Finally, in our physical galaxy the answer to your question is a big "yes!". The observed orbits of these stars [2] would be noticeably different if the central mass in the Milky Way's central parsec were anything but a black hole, and would be even more different if that central mass were not there at all.
- --
[1] Synge, J.L., _Gravitation. The General Theory_, ch. XI §8, "A five-point curvature detector".
[2] http://www.astro.ucla.edu/~ghezgroup/gc/animations.html and http://www.astro.ucla.edu/~ghezgroup/gc/blackhole.html
0.9 -> 0.99 -> 0.999 -> ... -> ?
0.1 -> 0.01 -> 0.001 -> ... -> ?
9/10 + 9/100 + 9/1000 + ... + 9/10^n + ...
1/10 + -9/100 + -9/1000 + -9/10000 + ...
But I just meant them as series, not necessarily as sums.
1 - 9/10 - 9/100 - ...
1 - ( 9/10 + 9/100 + ... )
0.00...1 = 1 - 0.999...
The decimal representation of a Real number has to be indexed by Natural numbers, i.e. every decimal digit[n] has a well-defined index n which is a Natural number. Infinity is not a Natural number, so 0.00...1 is not a Real number either.
EDIT: The number above seems well defined. It's lim n->inf (10^-n). That's zero.
The decimal representation of a Real number has to be indexed by Natural numbers, i.e. every decimal digit[n] has a well-defined index n which is a Natural number. Infinity is not a Natural number, so 0.00...1 is not a Real number either.
I will say though that if your explanation of 0.999... = 1.0 requires that you explain the distinction between countable and uncountable infinities, that's a big ask for most lay people.
I'm no expert, but countability doesn't depend on how the set is ordered. It depends on whether the elements can be placed in 1:1 correspondence with the integers. 1,0,0,... has a countable number of elements, and so does 0,0,0,...,1. They can be put in 1:1 correspondence with each other. This definition of countability is described in your link.
I meant to ask, does "countable representation" some kind of detailed definition that I can look at?
The limit of lim n->inf 10^-n is exactly 0, it is not 0.00...1.