I apologise for this, but I have numerous potential misunderstandings.
$\Bbb R^2$ is a metric space, since it is a subspace of the Euclidean space, $\Bbb R^n$
I can look at a set $E$ with elements $x_n$ where $x_n = \frac1n,n=1,2,\dots$ and look at the plot of $n$ vs $x_n$
A limit point is a point that has another point in any of its neighbourhoods.
This means that $(n,x_n)=(1,1)$ is an isolated point, taking an $r$ neighbourhood sufficiently small, lets say $r=.1$. Isolated points include $(2,\frac12),(3,\frac13)$ so on. Is this all correct reasoning so far?
Lastly this means my only limit points are at $(\to \infty, 0)$? [and hence not in $E$]