Re: Definition of real number ℝ --infinitesimal--

On 29/03/2024 04:29, olcott wrote:

x is said to be infinitesimal
if, and only if, |x| < 1/n for all integers n.
https://en.wikipedia.org/wiki/Hyperreal_number

That's for the hyperreals;  there's a clue in the URL.
There are no such "x" in R, by the Archimedean axiom.

0.999... specifies infinitesimally < 1.0

No it doesn't.  It specifies different things in different
number systems, which is why mathematicians don't use that notation
in contexts where there could be ambiguity.

and math guys have no way to say that so they
simply round up to 1.0

You've just referred to some "math guys" -- the proponents of
the hyperreals -- who say exactly what they mean by "infinitesimal".
You could equally have referred to the surreals [qv] where similar
statements are made and explained by "math guys".  Maths has moved
on over the past few centuries.  You and Wij need to move on with
the "math guys".
--
Andy Walker, Nottingham.
    Andy's music pages: www.cuboid.me.uk/andy/Music
    Composer of the day: www.cuboid.me.uk/andy/Music/Composers/Gottschalk