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I am having trouble rigorously proving/disproving the following:

Let $S \subset \mathbb{R}^n$ and let $F$ be the set of the frontier points of $S$. Is it true that the set of frontier points of $F$ is $F$ itself?

I believe I have the general idea, and tried to prove it using a contradiction (assume there is a frontier point not in $F$, therefore it must contain points in $S$ and be in F), but as said I am not certain as to how to fill in the "in-between," as far as definitions, etc go.

I appreciate all and any help. Thank you kindly!

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Hint: $F = \overline{S} \cap \overline{\mathbb{R}^n \setminus S}$ is the intersection of two closed sets. Hence $\overline{F} = \cdots$.

In addition, $\overline{A \cup B} = \overline{A} \cup \overline{B}$. (See, for instance, this answer.)

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