# Give example & prove there is a set with countable infinite set of accumulation point

1. Jun 15, 2011

### IntroAnalysis

1. The problem statement, all variables and given/known data

Give an example and prove there is a set with a countable infinite set of accumulations points.

2. Relevant equations
An example would be s = {k + 1/n l k element integers, n element natural numbers}

integers are countable infinitie a bijection exists with natural numbers

Def: Let S be a set of real numbers. A, element reals, is an accumulation point iff every neighborhood of A contains infinitely many elements of S.

Def: Let x element reals. Then a set Q, subset Reals, is called a neighborhood of x iff there exists epsilon > 0 such that (x -e, x + e) is a subset of Q.

3. The attempt at a solution

I've spent hours and don't know how to start to prove this. Would appreciate any help!
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Jun 15, 2011

### micromass

Staff Emeritus
Re: Give example & prove there is a set with countable infinite set of accumulation p

Do you know a set that's countable and dense in R? What about that set?

3. Jun 15, 2011

### IntroAnalysis

Re: Give example & prove there is a set with countable infinite set of accumulation p

Rationals are countable and dense in R. Still not sure where I am to take this.

4. Jun 15, 2011

### micromass

Staff Emeritus
Re: Give example & prove there is a set with countable infinite set of accumulation p

Aren't the rationals a set consisting of all accumulation points?

5. Jun 15, 2011

### Dick

Re: Give example & prove there is a set with countable infinite set of accumulation p

Are you thinking about rationals in Q? If you are thinking about rationals in R, then all points of R are accumulation points. That's not countable. What's wrong with {k+1/n}?

6. Jun 16, 2011

### micromass

Staff Emeritus
Re: Give example & prove there is a set with countable infinite set of accumulation p

Oh my, it appears I've been reading the questio entirely wrong

Yep {k+1/n | k,n naturals} are fine!