If you divide $10$ by $3$, you get $3.\overline{3}$ but $3.\overline{3}\times{3}=9.\overline{9}$ Does this make $9.\overline{9}=10$?
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2Yes, see the answer to the question why $0.999\cdots =1$ holds. – Peter Feb 07 '19 at 21:20
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The formula for the geometric series easily gives you the answer. – Peter Feb 07 '19 at 21:20
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Not in the $3$-adic metric. Try to prove convergence of $(9/1)+(9/10)+(9/100)+...$ to $10$ in that metric. You can't, because all the $3$-adic terms are multiples of $100_3=9$! – Oscar Lanzi Feb 07 '19 at 22:28
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It sure does!
One way to convince yourself is to look for a number in decimal representation that's greater than $9.\bar{9}$ but less than $10$.
John
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does that mean that $x.\overline{9}=y$ when y = the integer above x? – John Mobey Feb 09 '19 at 15:09
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