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It can't be $5$. And it can't be $4.\overline{9}$ because that equals $5$. It looks like there is no solution... but surely there must be?

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    $\begingroup$ Why must there be one? $\endgroup$ – Tobias Kildetoft May 20 '13 at 20:09
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    $\begingroup$ @Tom Not all subsets of the real numbers have a maximum, the set $\{x\in \Bbb R: x<5\}$ is one such instance. $\endgroup$ – Git Gud May 20 '13 at 20:10
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    $\begingroup$ Who says $x$ is a real number. I say $x$ is a number which is less than or equal to $4$ hence the greatest such $x$ is 4. Problem solved ;) $\endgroup$ – James S. Cook May 20 '13 at 20:11
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    $\begingroup$ If $x<5$ then $x<\frac{x+5}{2}<5$. So there is no greatest such $x$. $\endgroup$ – Thomas Andrews May 20 '13 at 20:13
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    $\begingroup$ What do you mean "must there still be a way to describe it?" There are lots of ways to describe things that don't exist. I can describe a moon made of green cheese, but that doesn't mean it exists. $\endgroup$ – Thomas Andrews May 20 '13 at 20:14
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There isn't one. Suppose there were; let's call it $y$, where $y<5$.

Let $\epsilon = 5 -y$, the difference between $y$ and 5. $\epsilon$ is positive, and so $0 < \frac\epsilon2 < \epsilon$, and then $y < y+\frac\epsilon2 < y+\epsilon = 5$, which shows that $y+\frac\epsilon2$ is even closer to 5 than $y$ was.

So there is no number that is closest to 5. Whatever $y$ you pick, however close it is, there is another number that is even closer.

Consider the analogous question: “$x < \infty$; what is the greatest value of $x$?” There is no such $x$.

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    $\begingroup$ To answer your analogous question, there is no such $x$ as a value, but $x = \infty-\frac1\infty$ would approach the limit quite nicely. $\endgroup$ – Gary S. Weaver May 21 '13 at 1:00
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    $\begingroup$ I brought that up only to illustrate the point that one can make up a set of properties that are not satisfied by any object. $\endgroup$ – MJD May 21 '13 at 1:12
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    $\begingroup$ Actually, if you wanted a simple way to describe this number, couldn't you just say $x = 5 - \frac1\infty$ ? $\endgroup$ – Darrel Hoffman May 21 '13 at 1:30
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    $\begingroup$ You can say whatever you like, but hardly anyone will understand what you mean. If you want a simple way to describe this particular collection of properties, then Tom's original description, the greatest $x$ that is less than 5, is perfectly clear. Your suggestion just obfuscates that. $\endgroup$ – MJD May 21 '13 at 1:38
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    $\begingroup$ @GaryS.Weaver What about $\infty - \frac1{2\infty}$? If that doesn't work, so many things are broken. $\endgroup$ – PyRulez May 30 '15 at 14:58
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The answer is $4$, assuming the domain of $x$ is $\Bbb Z$.

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    $\begingroup$ The original question mentioned 4.99999, so the intention should have been clear. $\endgroup$ – MJD May 20 '13 at 21:22
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    $\begingroup$ He said $4.999...$ which is in fact an integer. $\endgroup$ – PyRulez May 20 '13 at 21:23
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    $\begingroup$ It doesn't answer OP's question. $\endgroup$ – srijan May 20 '13 at 23:50
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    $\begingroup$ +1 because I always approve of picking an interpretation of a problem that renders it trivial. $\endgroup$ – Kyle Strand May 21 '13 at 3:56
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    $\begingroup$ The answer is 3, assuming the domain of x is {3}. $\endgroup$ – Plutor May 21 '13 at 11:48
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If $x<5$ then $2x<x+5$ so $x<\frac{x+5}{2}$. Similarly, $x<5$ means $x+5<5+5=10$ or $\frac{x+5}{2}<5$. So if $x<5$ we have $x<\frac{x+5}{2}<5$, and therefore there is a larger number, $\frac{x+5}{2}$ less than $5$.

Basically, the average of two different numbers must be strictly between those two numbers.

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