I'm practicing my proof-writing and was hoping you could let me know if this proof looks good. I would like to know if the proof is incorrect, if there are parts that are overly wordy/complicated, or if I'm missing some element of a proof that is helpful to see, if not strictly necessary.
The Prompt:
Is every point of every open set $E \subset \mathbb{R}^2$ a limit point of $E$? Answer the same question for closed sets in $\mathbb{R}^2$.
My Proof:
If $E$ is open, then for every $e \in E$ there is a neighborhood $N_r(e) \subset E$. Thus, every neighborhood of e contains some $x \in N_r(e)$ which is a member of $E$ and where $x \neq{e}$. Thus, every element of an open set is a limit point.
The same cannot be said for all closed sets. For example, $A = [0]$ is a closed set, but $0$ is not a limit point because there is no neighborhood of $0$ that contains an element of $A$ that is not $0$.
This is fine apart from a notational error in the argument for closed sets: even if you’ve already established that $0$ denotes the origin in $\Bbb R^2$, the closed singleton is $\{0\}$, not $[0]$. And if you’ve not established that convention, you want $\{\langle 0,0\rangle\}$.