Doubt reagarding denseness of a set in (0,1)

[ashok vardhan]

I have been doing a basic math course on Real analysis...I encountered with a problem which follows as" Prove that na(mod1) is dense in (0,1)..where a is an Irrational number , n>=1...

I tried to prove it using only basic principles...first of all i proved that above defined sequence is infinte..and also it is bounded...so by Bolzano-Weierstrass theorem it has a limit point in (0,1)..but to prove denseness i need to prove that for any given (a,b) a subset of (0,1) there is atleast one element of the sequence...Iam not getting how to figure out and link that limit point to that interval (a,b)..can anyone help me in this..?...It would be of great help...

 

[micromass]

OK, so you have proven that the sequence $na$ has a limit point in (0,1). Now, can you prove that for any $\varepsilon>0$, there is an element of the sequence such that $na<\varepsilon$ (that is: can you prove that 0 is a limit point).

 

[Dick]

I'll give you a starting hint. Can you show that since a is irrational, that no two points in the sequence na(mod 1) can ever be equal? Now partition the interval [0,1) in subintervals of size 1/K. Can you show one of the subintervals must contain at least two elements of na(mod 1)? It's basically the Pigeon Hole Principle.

 

[ashok vardhan]

sir, i tried to figure out that 0 is a limit point...but i am stuck with proving the fact that you mentioned i.e for any ε>0,there is ..an element of the sequence such that na<ε..I am able to understand why it should happen..but cannot prove it rigirously..Can u please help me out why such an integer "n" should exist..?...

 

[Dick]

sir, i tried to figure out that 0 is a limit point...but i am stuck with proving the fact that you mentioned i.e for any ε>0,there is ..an element of the sequence such that na<ε..I am able to understand why it should happen..but cannot prove it rigirously..Can u please help me out why such an integer "n" should exist..?...

Pick N large enough that 1/N<ε. Divide [0,1] into N equal parts. Can you argue that at least one of the parts contains at least two numbers of the form na(mod 1)?