user917
  • user917
Can someone explain this to me please?
Mathematics
  • Stacey Warren - Expert brainly.com
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schrodinger
  • schrodinger
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user917
  • user917
Given a real number x, and a natural number N>1. Consider the numbers : 0,x−⌊x⌋,2x−⌊2x⌋,…,Nx−⌊Nx⌋. Show that some pair of these numbers differs by at most 1/N.
anonymous
  • anonymous
Use the pigeon-hole principle
user917
  • user917
But what do they want from me? What should I proof?

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anonymous
  • anonymous
Prove that there exists a pair of numbers such that the difference between them is less or equal to than 1/N
user917
  • user917
oh ok thank you :D now I understand.. i'll think about it.. but how can the pigeon-hole principle help me?
anonymous
  • anonymous
I guess it is not exactly the pigeon-hole principle
anonymous
  • anonymous
The idea is to prove that the total distance between the numbers is greater than 1
user917
  • user917
no, the distance between the 2 numbers can't be greater than 1/N
anonymous
  • anonymous
Sorry, I meant to say that if the distance between all of the numbers is greater than 1/N, then the total distance is greater than 1. This is a contradiction, so there has to exist a pair of numbers with distance less than 1/N.
user917
  • user917
Why it is a contradiction?
anonymous
  • anonymous
\[0 \le x-\lfloor x \rfloor < 1\]
anonymous
  • anonymous
So all of the numbers are contained in an interval of length 1
user917
  • user917
Sorry but do you believe that I still can't get your point? what do you mean by "total distance" ?
anonymous
  • anonymous
Yeah, that is a bit ambiguous
user917
  • user917
why the total difference should be less than 1?
anonymous
  • anonymous
So if we suppose that all of the points are at a distance greater than 1/N from each other, 2 adjacent points on the number line must be separated by at least 1/N
anonymous
  • anonymous
Let's call P_0 the smallest point, P_1 the next largest, P_2 the next largest, et c.
anonymous
  • anonymous
The distance between point P_n and point P_n+x must be greater than x/N
anonymous
  • anonymous
The total distance is the distance between P_0 and P_N
anonymous
  • anonymous
This distance is clearly greater than 1
anonymous
  • anonymous
P_0 is 0
anonymous
  • anonymous
Therefore, P_N is greater than 1
anonymous
  • anonymous
This is a contradiction, as: \[x - \lfloor x \rfloor < 1\]
anonymous
  • anonymous
Therefore, our initial assumption, that the distance between any adjacent points is greater than 1/N, must be false
anonymous
  • anonymous
And that is the proof.
user917
  • user917
Thank you very much :D
anonymous
  • anonymous
You are welcome

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