I am self-learning Real Analysis from `Understanding Analysis` by Stephen Abbott. I learnt the trick of constructing a $\delta$-neighborhood of a random point $y \in (x - \epsilon, x+ \epsilon)$ to always reside in this $\epsilon$-neighborhood of $x$.

I tried to write a proof for the below assertion, and would like to ask if my proof checks out.

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> For any $S \subseteq \mathbf{R}$, if $\partial S$ denotes the boundary of $S$, prove that $\partial(\partial S) \subseteq \partial S$.
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***My Attempt***.

Let $x$ be an element of $\partial(\partial S)$.

By definition of the boundary of a set $S$, for every $\epsilon > 0$, the open interval $(x - \epsilon, x + \epsilon)$ contains at least one element of $\partial S$ and at least one element of $(\partial S)^C$.

We are interested to prove that, $\forall \epsilon>0$, the open interval $(x - \epsilon, x + \epsilon)$ contains at least one element of $S$ and at least one element of $S^C$.

Pick an arbitrary element $y \in (x - \epsilon, x + \epsilon)$, such that $y \in \partial S$. Then, for every $\delta > 0$, the open interval $(y - \delta, y + \delta)$ contains atleast one element of $S$ and atleast one element of $S^C$. If we choose,
\begin{align\*}
\delta = \delta\_0 = \min \left\{\frac{y - (x-\epsilon)}{2},\frac{(x + \epsilon) - y}{2}\right\}
\end{align\*}

Then, $(y - \delta\_0,y + \delta\_0) \subseteq (x - \epsilon, x + \epsilon)$.

So, we found a small $\delta\_0$-neighborhood, $(y - \delta\_0,y + \delta\_0)$ inside of $(x - \epsilon, x + \epsilon)$, which contains at least one element of $S$ and at least one element of $S^C$, for all $\epsilon>0$.

Consequently, we have shown that, $\forall \epsilon > 0$, $(x - \epsilon, x + \epsilon)$ contains at least one element of $S$ and at least one element of $S^C$.

So, $x \in \partial S$.

Hence, $\partial(\partial S) \subseteq \partial S$. This closes the proof.