Here's Theorem 2.27 (a) in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition:
If $X$ is a metric space and $E \subset X$, then $\overline{E}$ is closed.
Now here's Rudin's proof:
If $p \in X$ and $p \not\in \overline{E}$ then $p$ is neither a point of $E$ nor a limit point of $E$. Hence $p$ has a neighborhood which does not intersect $E$. The complement of $\overline{E}$ is therefore open. Hence $\overline{E}$ is closed.
Is the above proof good enough, especially at the level Rudin is intended for?
Now here's the proof I propose:
If $p \in X$ and $p \not\in \overline{E}$ then $p$ is neither a point of $E$ nor a limit point of $E$. Hence $p$ has a neighborhood which does not intersect $E$. Let $N_\epsilon (p)$ be this neighborhood.
Now we show that no point of $N_\epsilon (p)$ can be in $\overline{E}$. Let $q \in N_\epsilon (p)$. Then $d(q,p) < \epsilon$, where $d$ denotes the metric on $X$.
Let $\delta \colon= \epsilon - d(p,q)$. Then $0 < \delta < \epsilon$. Now if $a \in N_\delta (q)$, then $d(a,q) < \delta = \epsilon - d(q,p)$, which implies that $$d(a,p) \leq d(a,q) + d(q,p) < \epsilon,$$ and so $a \in N_\epsilon (p)$.
Thus we have shown that $N_\delta (q) \subset N_\epsilon (p)$. Since $ N_\epsilon (p) \cap E = \emptyset$, we have $N_\delta (q) \cap E = \emptyset$ as well. That is, the point $q$ has a neighborhood --- namely $N_\delta (q)$ --- which does not intersect $E$ at all. So $q \not\in \overline{E}$.
But $q$ was an arbitrary point in $N_\epsilon (p)$. So $N_\epsilon (p) \subset \left( \overline{E} \right)^c$.
But $p$ was an arbitrary point in $\left( \overline{E} \right)^c$. Thus, we can conclude that every point of $\left( \overline{E} \right)^c$ is an interior point. Hence $\left( \overline{E} \right)^c$ is open.
Now is my proof any better than Rudin's? Are there any extra advantages to be had from inclusion or exclusion of extra details?
I would say that your proof is better. I would assume that he missed a detail, and left out a proof that if it holds for $E$ then it holds for $\overline{E}$. Certainly, if I were grading a course I would mark his proof as incomplete - even in a course not for first or second years. Especially since his book is a standard introductory text for first and second years, I think this oversight is problematic.
As a commenter notes, authors have a habit of increasing the details they omit as time goes on, out of a combination of laziness, a desire to save space, and a desire to write less, but at chapter 2 of an intro book rigor should be the standard.