I want to prove the following theorem:
A subgraph $H$ of $G$ is connected if and only if it contains an edge from every cut of $G$ that separates two of its vertices.
My attempt:
Let subgraph $H$ of $G$ be connected. Suppose that $(V_1,V_2)$ is an arbitrary cut. If $V(H)$ has an intersection with both $V_1$ and $V_2$, then we done. So we can assume that $V_1$ contains $V(H)$.