Confusion about the boundary of connected components

Let $x \in \mathrm{cl}(C)$. Then, suppose that $x \in \mathrm{int}(X)$. Since $\mathbb{R}^n$ is locally connected, there is a connected set $V \subset X$ which is a neighborhood of $x$.

Since $C$ is connected and $V$ intersects $C$, $C \cup V \subset X$ is connected (why?). Therefore, $V \subset C$. That is, $x \in \mathrm{int}(C)$.

That is, for any $x \in \mathrm{cl}(C)$, $$ x \in \mathrm{int}(X) \Rightarrow x \in \mathrm{int}(C). $$ Therefore, $$ \partial C = \mathrm{cl}(C) \setminus \mathrm{int}(C) \subset \mathrm{cl}(C) \setminus \mathrm{int}(X) \subset \mathrm{cl}(X) \setminus \mathrm{int}(X) = \partial X. $$

Notice that this proof works for any locally connected space in place of $\mathbb{R}^n$.

Edit: Proof made much much simpler (and correct :-P).