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I want to prove that $1,999\dots$ Is an element of $\mathbb{Z}$. Here is my try :

$x = 1,9999\dots \\ 10x = 19,9999\dots \\ 10x - x = 18 \\ 9x = 18 \\ x = 18/9 \\ x = 2 $

So $x \in \mathbb{Z}$

I know something is wrong but where ?

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  • $\begingroup$ How do you define $1.999 \ldots$ ? $\endgroup$
    – Martin R
    Mar 1, 2018 at 20:19
  • $\begingroup$ 1 with a infinity of 9999 after the comma $\endgroup$ Mar 1, 2018 at 20:22
  • $\begingroup$ Why do you think something is wrong with what you wrote? $\endgroup$ Mar 1, 2018 at 20:29
  • $\begingroup$ @Mathematician42 because $1,9\dots \neq 2$... $\endgroup$ Mar 1, 2018 at 20:32
  • $\begingroup$ Actually you have just proved that they are indeed equal. Well done. And no, there is nothing wrong with that. $\endgroup$
    – Joffan
    Mar 1, 2018 at 20:41

1 Answer 1

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$$x=1+9\sum_{i=1}^{+\infty}10^{-i} $$

$$=1+9\sum_{i=1}^{+\infty}(\frac {1}{10})^i$$

$$=1+9\frac {1}{10}\frac {1}{1-\frac {1}{10}}$$

$$=1+9\frac{1}{10}\frac {10}{9}=2$$

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