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Reals are just point cores of dressed Cauchy sequences of naturals (think of it as a continually constructed set of narrowing intervals "homing in" on the real being constructed). The intervals shrink at the same rate generally.
1!=0.999 iff we can find an n, such that the intervals no longer overlap at that n. This would imply a layer of absolute infinite thinness has to exist, and so we have reached a contradiction as it would have to have a width smaller than every positive real (there is no smallest real >0).
Therefore 0.999...=1.
However, we can argue that 1 is not identity to 0.999... quite easily as they are not the same thing.
This does argue that this only works in an extensional setting (which is the norm for most mathematics).
Easiest way to prove it:
1 = 3/3 = 1/3 * 3 = 0.333... * 3 = 0.999...
Ehh, completed infinities give me wind...
extensional setting
Very good answer. The only missing thing is what extensionality is.
Thanks for the bedtime reading!
I mostly deal with foundations of analysis, so this could be handy.