Let $X$ be a topological space. Let $A \subset X$. The point $x$ is a limit point of $A$ iff every neighborhood of $x$ contains a point $a$ of $A$ not equal to $x$.
I am thinking about the following question: If $x$ is a limit point of $A$ does it hold that there exists a sequence $x_n $ in $A$ converging to $x$?
Please can you tell me if my thoughts are correct: If $x$ has infinitely many neighborhoods then there are infinitely many different points $x_n \in A$ and for larger $n$ they are nearer $x$ therefore $x_n \to x$ and it is true. If $x$ only has finitely many neighborhoods then it is also true: take the smallest neighborhood and the point $x_n = a \in A$ in that neighborhood and then the constant sequence $a$ converges to $x$ (even though $x\neq a$).
Best Answer
The property that you're describing does not hold in every topological space. If the limit points can be characterized by limits of convergent sequences, then one calls $X$ a sequential space (Edit: as pointed out in the comment, there are some subtleties involved in the exact definition, see here).
Every first-countable space is sequential. Most "real-world" topological spaces are first-countable.
But one can also make up simple examples of nonsequential spaces. Let $X$ be any uncountable set and define a topology on it by calling every subset of $X$ whose complement is countable open. Additionally, call the empty set open. This is also called the cocountable topology. That space answers your question in the negative.
The problem with sequences is that they by nature countable, because they are indexed by natural numbers. You can however generalize the concept of sequences to nets (Moore-Smith sequences), where you can use any set as index set instead. In the language of nets it is really true that every limit point is the limit of a convergent net.
Edit: Let us prove that the cocountable topology is not sequential.
Proof: Define $Y=\{x_n\,:\,n\in\mathbb{N},\,x_n\not=x\}\subset X$. Because $Y$ is countable, its complement $U=X\backslash Y$ is open. Also, $x\in U$ because $x\not\in Y$. So $U$ is a neighborhood of $x$. By definition of convergence, there is an $N$ such that $x_n\in U$ for all $n\ge N$. But that means $x_n\not\in Y$. That is, $x_n=x$ for $n\ge N$. $\square$
That means, every convergent sequence is eventually constant. So if we consider an arbitrary set $A\subset X$, then we do not gain any new points by considering limits of sequences in $A$. Every set in $X$ is sequentially closed.