Proof of existing path on Depth-First-Search spanning tree

Question

Let $G$ be an undirected connected graph, and $T$ the directed spanning tree of $G$, which I got by performing a DFS on $G$. If $H$ is a complete subgraph of $G$, how can I proof that there a path in $T$ exists which passes all nodes in $H$? And is it possible that that path passes other nodes, which are not in $H$?