I'm trying to solve the following problem:
We call a point $x$ an accumulation point for a sequence $(x_n)$ in a topological space $X$ if every neighbourhood $U$ of $x$ contains infinitely many elements of the sequence. Show that the set of accumulation points equals $\bigcap_n F_n$ where $F_n = \overline{ \{ x_k \vert k \geq n \} }$. Use this to show that the set of accumulation points is closed and nonempty if $X$ is compact.
I'm stuck on the first one:
Show that the set of accumulation points equals $\bigcap_n F_n$ where $F_n = \overline{ \{ x_k \vert k \geq n \} }$.
It seems intuitively obvious, but I don't know how to show it.
Any hints on where to begin greatly appreciated!