This is from P.580 of Serge Lang's undergraduate analysis (2nd edition).
$\textbf {Proposition 2.3.}$ Let $A$ be an admissible set in $\mathbb R^n$ and assume that its closure $\bar{A}$ is contained in an open set $U$. Let $f \colon U \to \mathbb R^n$ be a $C^1$ map, which is $C^1$-invertible on the interior of $A$. Then $f(A)$ is admissible and $$\partial f(A)\subset f(\partial A).$$
Proof. Let $A^o$ be the interior of $A$, that is the set of points of $A$ which are not boundary points of $A$. Then $A^o$ is open, so is $f(A^o)$, and $f$ yields a $C^1$-invertible map between $A^o$ and $f(A^o)$. We have $$\bar{A}=A^o \cup \partial A,$$ and $\partial A = \partial \bar{A}$, whence $$f(A^o) \subset f(A) \subset f(\bar{A}) = f(A^o) \cup f(\partial A).$$ This shows that $\partial f(A)\subset f(\partial A),$ and that $\partial f(A)$ is negligible by Proposition 2.1, thus proving Proposition 2.3.
I have difficulty in understanding the proof, hope that someone here can enlighten me.
My questions are:
i) On line 4 of the proof, why is it true that $\partial A = \partial \bar{A}$?
ii) Why do we need this fact in this proof?
I tried to ask the questions in MSE but received no reply. Thank you in advance for taking time to answer my question.
PS: According to Lang, a set is said to be admissible if it is bounded and its boundary is a negligible set. This is usually called a Jordan domain. A negligible set is merely a set of Jordan content zero.