I am trying to get a better understanding of accumulation points defined in the following way: A point $a$ is an accumulation point for $E = {x_1,x_2,x_3,...}$ if and only if
$\forall r>0, E \cap B_{r}(a)\setminus \{a\}$ is non-empty. Let $d(x,y)$ be the distance between two points $x$ and $y$ in E. Now suppose that $L$ is the set of all accumulation points of $E$ and define $B_r(E) = \{x \in E such that d(x,L)<r\}$. Is it necessarily true that $\forall r>0$, there exists $n_r \in mathbb{N}$ such that $\forall n>n_r, x_n \in B_r(E)$? Intuitively this seems to be true, but I haven't been abl…