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

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In summary, the problem at hand is to prove that the sequence na (mod 1) is dense in the interval (0,1), where a is an irrational number and n is greater than or equal to 1. The first step is to prove that the sequence is infinite and bounded, using the Bolzano-Weierstrass theorem. Then, to prove denseness, the challenge is to show that for any given interval (a,b) within (0,1), there is at least one element of the sequence present. The conversation then moves on to discussing how to show that 0 is a limit point and how to prove that for any ε>0, there is an element of the sequence such that na<ε
  • #1
ashok vardhan
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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...
 
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  • #2
OK, so you have proven that the sequence [itex]na[/itex] has a limit point in (0,1). Now, can you prove that for any [itex]\varepsilon>0[/itex], there is an element of the sequence such that [itex]na<\varepsilon[/itex] (that is: can you prove that 0 is a limit point).
 
  • #3
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.
 
  • #4
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..?...
 
  • #5
ashok vardhan said:
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)?
 

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

1. What is the definition of a dense set?

A set is considered dense in a given space if every point in that space can be arbitrarily well-approximated by elements in the set. In other words, there are no gaps or holes in the set and it is "closely packed" in the space.

2. How do you determine if a set is dense in (0,1)?

To determine if a set is dense in (0,1), you would need to show that for any given point x in (0,1), there exists a sequence of elements in the set that converges to x. This means that any neighborhood around x will contain elements from the set, no matter how small the neighborhood is.

3. What is the significance of a set being dense in (0,1)?

If a set is dense in (0,1), it means that the set is "spread out" across the entire interval, and there are no gaps or missing points. This can have implications for various mathematical and scientific applications, such as approximation and continuity.

4. Can a set be dense in (0,1) but not dense in the real numbers?

Yes, it is possible for a set to be dense in (0,1) but not dense in the real numbers. This means that the set is "closely packed" in the interval (0,1), but there may be gaps or missing points when considering the entire set of real numbers.

5. Are there any specific examples of dense sets in (0,1)?

Yes, there are many specific examples of dense sets in (0,1). These include the set of rational numbers, the set of irrational numbers, and the set of all algebraic numbers. Additionally, any infinite subset of (0,1) can be dense in (0,1).

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