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:

1. The frontier $\partial S$ of a subset $S$ is closed.
2. $ \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 $\partial(\partial S) \subseteq \partial S$.