1. Limit point Definition

hey all,
I read the following definition of a limit in some lectures notes:
"Suppose that A⊆X (in a metric space). The point $\displaystyle x_0 \in X$ is a limit point of A if for every neighborhood $\displaystyle U(x_0, \epsilon)$, of xo, the set $\displaystyle U(x_0, \epsilon)$ is an infinite set."
Well, the definition I know is: $\displaystyle x_0$ is a limit point of a set A if every deleted neighborhood of $\displaystyle x_0$ intersects A in at least one point.
Can someone explain to me the first definition? I can't make the link between the two definitions
thanx

2. If $\displaystyle U(x_0;\epsilon)$ infinitely many points then then $\displaystyle U(x_0;\epsilon)\setminus\{x_0\}$ contains at least one point.

What don't you understand?

3. OK, but any neighborhood $\displaystyle U(x_0;\epsilon)\setminus\{x_0\}$ contains at least one point anyway. It's a neighborhood. They don't mention intersection with A here! That's what I'm not getting.
thanx

4. Well I think the it should be understood that it is infinitely many points of A.