Equivalence of closure of a sequence

Question

Let be $(X, || \cdot ||)$ a normed space and $A:=\{ a_n \}_{n \in \mathbb{N}} \subset X$ a sequence, then: \begin{equation*} cl \{ a_n \}_{n \in \mathbb{N}} = A \cup P \end{equation*} Where cl denotes the closure and $P = \{ x \in X: \hspace{.1cm} (B_\epsilon(x)\setminus \{ x \})\cap A \neq \varnothing \hspace{.1cm} \forall \epsilon >0 \}$ i. e. the accumulation points set. I have tried to do the proof but can not achieve it, the contention $A \cup P \subset cl \{ a_n \}_{n \in \mathbb{N}}$ is immediate.