Question: $X$ is connected and $X\subset Y\subset\bar X$, prove that $Y$ is connected.
This is one of my midterm questions this morning. I couldn't figure it out. But now I came up with this proof, so if anyone could check and see if it makes sense, I would really appreciate it (although that's not gonna change my grade).
Proof: For the purpose of arguing by contradiction, we assume that $Y$ is disconnected. Then we have nonempty, open sets $U,V$ such that $U \cup V=Y$ and $U \cap V=\emptyset$.
Since $X \subset Y$, then it's obvious that we either have $X \subset U$, or $X \subset V$. Without losing generality, we assume that $X\subset U$. Then $\bar X \subset \bar U$, then $Y \subset \bar U$, then $V \subset \bar U$.
Consider some $v \in V$. There exits some $\epsilon$ such that $B_\epsilon (x)\subset V$. At the same time, since $x\in V \subset \bar U$, $B_\epsilon(x) \cap U \neq \emptyset$. This is a contradiction since $U \cap V =\emptyset$. So $Y$ is connected.