I studied a definition,and I didn´t find it in anyother book (but those I use). It´s like a point of closure for sequences.We call $a$ "value of closure" of $(x_n)$ when $a$ is the limit of a subsequence of $(x_n)$.
The question is:
For a real number $a$ be a "value of closure" is necessary and sufficient that $\forall \epsilon >0$ and $\forall k \in \mathbb{N}$ given ,there is $n > k$ such that $|x_n -a|< \epsilon$.
I could do the first part ($a \Rightarrow |x_n - a|<\epsilon$) but not the $\Leftarrow$.
Thanks for any help!