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I know that mathematics says that $0.99999....=1$. but some say that logic says that $0.99999....<1$. what I want to know, is weather or not there is logical proof (proof without formulas, and that relies solidly on common sense ) that $0.99999....$ must be equal to $1$.

I have heard great arguments for both sides, but they all seem to end with logic and math disagreeing. I want answers that are insightful, and make math and logic agree with each other.

OK a lot of people are asking what the logical proof is, well a few of the ones that I have heard of so far are: If you move a ninth of distance to your destination then a ninth of whats left, then a ninth of whats left after that, and so on, and so on, you will never get there. Another logical question that comes up is: How can an infinite sum be equal to any non fractional whole number, isn't it never ending? can you help me explain why these ideas are flawed without using math?

Also if you'r wondering what I mean when say logical proof, I mean proof by logical deduction with as little use of mathematical formulas as possible, in other words I want proof using common sense. all of the other proofs I have seen are great and fallow the laws of mathematics perfectly, but their are some people that agree with their own common sense more than math. I want an answer I can give to them. so the less formulas the better, try to rely on firm logic.

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    $\begingroup$ The first question is, what do you even mean by $0.9999....$? $\endgroup$ – user160738 Nov 8 '16 at 18:16
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    $\begingroup$ If the dozens of answers already on this site don't satisfy you, then there should be no reason to believe that any replies to your question will be any better. Any “logical” argument that $0.999… < 1$ must turn on a vague or incorrect understanding of the meaning of “$0.999…$”. Once this is nailed down precisely, the question is easy to resolve. $\endgroup$ – MJD Nov 8 '16 at 18:17
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    $\begingroup$ Do a site search for "0.999" and you should get plenty of insights. In the meantime, I'm going to vote to close this as a duplicate of this specific question (the eleventh question ever posted to the site!): "Is it true that $0.999999999…=1$?" $\endgroup$ – Blue Nov 8 '16 at 18:17
  • $\begingroup$ Do you have any evidence whatsoever that "logic" and "mathematics" give different values of $0.9999\ldots$, other than the fact that you say they do? $\endgroup$ – David K Nov 8 '16 at 18:25
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    $\begingroup$ There is no debate between people in advanced mathematics on this subject, so you're not going to get any representations for 0.99999... < 1, simply because it isn't. $\endgroup$ – Kaynex Nov 8 '16 at 18:39
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You have $0.999\cdots=1$ because of this:

Consider the sequence $$u_n:=\sum_{k=1}^n 9\cdot 10^{-k}.$$

You have

$$0.999\cdots=\lim_{n\to\infty} u_n=\sum_{k=1}^{\infty} 9\cdot 10^{-k}=9\frac 1{10}\frac{1}{1-\frac 1{10}}=1.$$

You will never have $0.999\cdots<1$.

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