Let $(X,d)$ a metric space. Let $F$ and $A$ two subsets of $X$ such that $A\cap F=\emptyset$ and $F$ is closed. Suppose that for any converging sequence $\{u_{n}\}_{n\in \mathbb{N}}\subset A$, we have $\lim x_{n} \in A\cup F$.
question: Does it follow that $A\cup F$ is a closed subset of $X$ ?
Hint Let $x_n \in A \cup F$ be convergent to some $x \in X$. You need to show that $x \in A \cup F$.
Split the problem in two cases:
Case 1: There are infinitely many $x_n$ in $A$. Then, they form a subsequence of $x_n$ which is in $A$ and converge to $x$.
Case 2: There are finitely many $x_n$ in $A$. Then, you can find a subsequence of $x_n$ which is in $F$ and converge to $x$.