accumulation point proof

[autre]

I need to show that if every neighborhood of some x\in A for some A\subseteq\mathbb{R} contains infinitely many points of A, then x is an accumulation point of A.

So far, I have:

Let A\subseteq\mathbb{R}. I want to show that if every neighborhood of x\in A has infintely many points of A, there exists a y\in\mathbb{R} such that y\in((x-\epsilon,x+\epsilon)\bigcap A\{x}).

Am I on the right track?

[mathman]

Looks good.

[autre]

I'm having some trouble finding that y.

Let A⊆R. Define the neighborhood of x∈A as (x−ϵ,x+ϵ) \forall\epsilon>0. Since x∈A, (x−ϵ,x+ϵ)⋂A is not empty. Let y∈(x−ϵ,x+ϵ)⋂A...how do I show that y\not=x?

Thanks!

[micromass]

I'm having some trouble finding that y.

Let A⊆R. Define the neighborhood of x∈A as (x−ϵ,x+ϵ) \forall\epsilon>0. Since x∈A, (x−ϵ,x+ϵ)⋂A is not empty. Let y∈(x−ϵ,x+ϵ)⋂A...how do I show that y\not=x?

You can't in general. It could very well be that you chose y=x. Your choice doesn't disallow this. However, you know that (x−ϵ,x+ϵ)⋂A is infinite. So it doesn't only contain x, does it??

[autre]

However, you know that (x−ϵ,x+ϵ)⋂A is infinite.

Right, but don't I have to prove that this intersection is infinite? I'm trying to show that x is an accumulation point, but (x−ϵ,x+ϵ)⋂A being infinite presupposes x being an accumulation point, right?

[gb7nash]

if every neighborhood of some x\in A for some A\subseteq\mathbb{R} contains infinitely many points of A, then x is an accumulation point of A.


I might be missing something, but you're trying to prove the definition of an accumulation point (which you can't do). The assumption is that every neighborhood of some x in A contains infinitely many points of A. Therefore, it is an accumulation point by definition.

[micromass]

Right, but don't I have to prove that this intersection is infinite? I'm trying to show that x is an accumulation point, but (x−ϵ,x+ϵ)⋂A being infinite presupposes x being an accumulation point, right?

Isn't that given?? The first line of the post is


if every neighborhood of some x∈A for some A⊆R contains infinitely many points of A


So you are given that the neigbourhood (x−ϵ,x+ϵ) contains infinitely many points of A.

Or am I totally misunderstanding your question?

[autre]

Or am I totally misunderstanding your question?


No, you're right. the proof seemed too simple so I thought I was missing something. Makes sense now.