I am taking a course that is using Fitzpatrick's Advanced Calculus and we are currently discussing limit points. I know that there are more general topological ways of describing limit points but I would like to restrict our discussion to the following definition.

Definition: For a subset $D$ of the real numbers, the number $x\_0$ is called a limit point of $D$ if there is a sequence in D\ {$x\_0$} that converges to $x\_0$.

My question is actually not directly from the textbook, but in my search to understand the topic I came across some lecture notes from another course at a different university, where the following example was given

1 and 3 are limit points of $D = (1,3)$ but 0.5 is not. 

I understand why 1 and 3 are limit points, because you can find a sequence in (1,3) that converges to both points even though those points are not in the set. 

But I don't understand why 0.5 is not called a limit point. Couldn't I find some sequence in (1,3) that converges to 0.5 without actually reaching 0.5? I can't think of one off the top of my head though.