Can someone explain this to me please?

Question

user917 · Answer

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 · Answer

Use the pigeon-hole principle

user917 · Answer

But what do they want from me? What should I proof?

anonymous · Answer

Prove that there exists a pair of numbers such that the difference between them is less or equal to than 1/N

user917 · Answer

oh ok thank you :D now I understand.. i'll think about it.. but how can the pigeon-hole principle help me?

anonymous · Answer

I guess it is not exactly the pigeon-hole principle

anonymous · Answer

The idea is to prove that the total distance between the numbers is greater than 1

user917 · Answer

no, the distance between the 2 numbers can't be greater than 1/N

anonymous · Answer

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 · Answer

Why it is a contradiction?

anonymous · Answer

$$0 \le x-\lfloor x \rfloor < 1$$

anonymous · Answer

So all of the numbers are contained in an interval of length 1

user917 · Answer

Sorry but do you believe that I still can't get your point? what do you mean by "total distance" ?

anonymous · Answer

Yeah, that is a bit ambiguous

user917 · Answer

why the total difference should be less than 1?

anonymous · Answer

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 · Answer

Let's call P_0 the smallest point, P_1 the next largest, P_2 the next largest, et c.

anonymous · Answer

The distance between point P_n and point P_n+x must be greater than x/N

anonymous · Answer

The total distance is the distance between P_0 and P_N

anonymous · Answer

This distance is clearly greater than 1

anonymous · Answer

P_0 is 0

anonymous · Answer

Therefore, P_N is greater than 1

anonymous · Answer

This is a contradiction, as:
$$x - \lfloor x \rfloor < 1$$

anonymous · Answer

Therefore, our initial assumption, that the distance between any adjacent points is greater than 1/N, must be false

anonymous · Answer

And that is the proof.

user917 · Answer

Thank you very much :D

anonymous · Answer

You are welcome