limit point and limit of a sequence

Question

In a metric space $(E,d)$ with $A\subseteq E$, if a sequence in $A$ satisfies $a_n\rightarrow a $, then this point $a\in E$ is a limit point of $A$?

I know that limit points are defined for sets and limits for sequences, but a sequence is also a set, right?

Answer 1

Not necessarily. Suppose that $A=\{a\}$. Then your condition holds (just take $a_n=a$ for each $n\in\mathbb N$), but $a$ is not a limit point of $\{a\}$.