Proving Countable Infinite Accumulation Points in a Set

[IntroAnalysis]

Homework Statement

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

Homework 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.

The Attempt at a Solution

I've spent hours and don't know how to start to prove this. Would appreciate any help!

 

[micromass]

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

 

[IntroAnalysis]

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

 

[micromass]

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

 

[Dick]

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

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}?

 

[micromass]

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

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