Let $G = (V,E)$ be a graph and let $H_1 = (V_1,E_1)$ and $H_2 = (V_2,E_2)$ be two connected subgraphs of $G$ that have at least one node in common. Prove that the graph $H = H_1\cup H_2 = (V_1\cup V_2,E_1\cup E_2)$ is connected.
Pick a random node $V_i$ in $H_1 \cup H_2$. Since $H_1$ is connected, I can reach from $V_i$ to the common node of $H_1$ and $H_2$. Since $H_2$ is connected, I can reach from $V_i$ to nodes in $H_2$. Thus, $H_1\cup H_2$ is connected. Is this the correct logic? How do I write it out formally?