Question: Find the limit of $A={\{(\dfrac{\theta-1}{\theta}}, \theta)|\theta=1,2,3,\dots\}$?
Here, $\left(\dfrac{\theta-1}{\theta},\theta\right)$ is a point in $\mathbb R^2$ expressed in polar coordinates.
Incomplete answer: At first look, I can say there is no limit point for such a sequence since it goes and goes on and this set of points spirals out toward the circle $S^1$ ($r=1$), but no obvious point is different than others on $S^1$ to be the limit point (that is the point where this set 'accumulates'). So it doesn't converge at all.
But, on the other hand, the disk is compact and therefore limit point compact. This implies that every infinite subset has a limit point. Since $A$ is an infinite subset, so it must have a limit point!
Where does this contradiction come from? And what is the limit of the sequence $A$?
Many thanks.
EDIT - According to the def. of the book: A topological space X is limit point compact if every infinite subset of X has a limit point.
$A$ does indeed have a limit point $-$ in fact an infinite number of them. Every point on the unit circle is a limit point.