Procedure of finding accumulation points of set

Question

For example, there is a set $R$ and I want to find all accumulation point(or to show, that they don't exist) of $Z$(integers)

I am familiar with concepts of open/closed balls, neighbourhood, $\epsilon$ - neighbourhood, interior pointm open/closed sets, accumulation points.

But I don't know the mechanism of proving(finding/showing), that these points exists or not for subset.

Can you provide me a procedure, describing each step?

Answer 1

Think about an extreme example. Let A = {1} $\subset \mathbb{R}$. What are the limit points of A. Extend this to any finite set of integers, then to all integers

EDIT: proof by contradiction

suppose x $\in \mathbb{R}$ and x is an accumulation point of $\mathbb{Z}$. Then for all $\epsilon \gt 0$, there exists a point of p $\in \mathbb{Z}$ such that p $\not=$ x and |x - p| $\lt \epsilon$...

find an $\epsilon$ that does not work