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Is $0.\overline{0}1 \in \mathbb{R}$? If so, is $0.\overline{0}2>0.\overline{0}1$ and $0.\overline{0}1 \neq 0$? If not, is there a way to define the smallest Number that is not $0$?

LostReminder
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    no it is not, take any number $x$ then $x/{2}$ is closer to $0$ – Tortar Jul 23 '22 at 20:36
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    The symbol $0.\bar{0}1$ does not exist as a real number. The closest concept is the idea of infinitesimals in non-standard analysis. – Cameron L. Williams Jul 23 '22 at 20:36
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    You have to define what you mean by the expression ${0.\overline{0}1}$ in the first place, and there does not seem to be a sensible way to do so. At first glance, the expression hints at a smallest positive real number, which is obviously nonsense. And so I cannot imagine there being a sensible way of defining it – Riemann'sPointyNose Jul 23 '22 at 20:38

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Define a number, say $r$, that is "the smallest number not 0" then $r/2\in \mathbb{R}$ and $r/2>0$, a contradiction, so, no, there isn't any way to define the smallest number not 0.

Suzu Hirose
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