Suppose $M⊆\mathbb{R}^n$ is connected and $M⊆N⊆\overline{M}$. Prove that N is connected. I have gotten to a point where I need to show there exists an element in $M\cap V$$\neq$$\emptyset$ and $M\cap U$$\neq$$\emptyset$, where V and U are relatively open sets in $R^n$. A friend helped me earlier and said $\overline{M}=\partial{M}\cup M$. This fact hasn't been defined in the book as of yet though, so I need help going in another direction.

If $\overline{M}$ is disconnected, say $\overline{M} \subseteq U\cup V$ where $U, V$ are disjoint open sets with $\overline{M}\cap U, \overline{M}\cap V$ non empty, then $M\subseteq U\cup V$. Because $M$ is connected, it must be contained in $U$ or $V$, say $M\subseteq U$.
Then, $V\cap \overline{M} = \emptyset$ because every point in $V$ has a neighbourhood, namely $V$, that is disjoint from $M$, and this is a contradiction.
Hence $\overline{M}$, and similarly any $M\subseteq N\subseteq\overline{M}$, is connected.
