For any $S \subseteq \mathbf{R}$, if $\partial S$ denotes the boundary of $S$, prove that $\partial(\partial S) \subseteq \partial S$.
What you did looks good to me.
Below another proof, that you can use depending on what you already proved.
If you know that:
- The frontier $\partial S$ of a subset $S$ is closed.
- $ \partial A \subseteq B$ whenever $ A \subseteq B$ and $B$ is closed.
Then is it almost immediate as $\partial S \subseteq \partial S$ and $\partial S$ is closed. Therefore $\partial(\partial S) \subseteq \partial S$.