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Whether or not a specific construction is well defined depends on the system you are using. This representation is certainly well definable. I think your objec

• aw1621107 6y ago
  > This representation is certainly well definable. Do you mind defining it more formally for me then or pointing me to explanations/systems that would permit such a definition? I'm not really sure what such a formulation might look like, but then again I'm not nearly well-acquainted enough with mathematical topics past what is commonly taught. Does it involve the hyperreals, surreals, or one of the other systems beyond the "standard" reals as mentioned by other comments in the thread? > I think your objection here is in the same vein as those who object to the notion that 0.999... = 1.0 > For most people, the concept of an infinite representation is not well defined. I think (or at least hope) it's a bit more nuanced than that. I understand that some reals with infinite decimal representations can be well defined as an infinite series, and that defining 0.999... using such a series allows other manipulations to be done to complete the proof. However, adding the concept of an "end" to said infinite series kind of breaks that understanding. The translation to an infinite sum no longer seems to hold, so I'm at a bit of a loss. It's also somewhat counterintuitive to have an "end" to infinity, but as the rest of the thread shows intuition isn't always reliable for this kind of thing, especially for those who aren't particularly familiar with more detailed bits of math.