0.(9) ≠ 1

...

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en.wikipedia.org/wiki/Least-upper-bound_property
en.wikipedia.org/wiki/0.999...
en.wikipedia.org/wiki/Talk:0.999...
en.wikipedia.org/wiki/Hilbert's_paradox_of_the_Grand_Hotel),
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wtf is 0.(9)?

[math]
\displaystyle \lim_{n\to \infty} \sum_{k=1}^n \frac{9}{10^k} = 1
[/math]
btw

You just wrote "0.(9) = 1".

Not equals, only converges to one.

Then: "0.(9) -> 1"

n can not reach infinity

[math] \displaystyle
1 = \frac {3}{3} = 3 \cdot \frac {1}{3} = 3 \cdot 0. \bar{3} = 0. \bar{9}
[/math]

so not converges to one

the problem with this is that 3 * 0.333... =/= 0.999...