Say $x$ is a limit point of a set $A$.
That means that for any $r > 0$, $B_r(x)$ contains a point of $A$ other than $x$ itself.
Or I often see it stated as $B_r(x) \cap A \neq \varnothing$ and these statements are taken to be equivalent.
But they don't seem equivalent to me. Say I have a set $A = \{2\} \cup [4,6]$ in $\mathbb{R}$.
So if we consider the point $2$, the first statement above says that $2$ will not be a limit point. However if we take the $B_r(x) \cap A \neq \varnothing$, $2$ will be a limit point because that open ball contains $2$ itself.
The $B_r(x) \cap A \neq \varnothing$ definition seems invalid as it doesn't take account of the fact that we only want to consider points in the ball other than $x$. So have I missed something or is $B_r(x) \cap A \neq \varnothing$ not a valid statement when it comes to defining a limit point of a set?