Connected Subgraph of Tree is Tree

Theorem
Let $$T$$ be a tree.

Let $$S$$ be a subgraph of $$T$$ such that $$S$$ is connected.

Then $$S$$ is also a tree.

Proof
Follows directly from the fact that by definition, $$T$$ has no circuits.

As $$T$$ has no circuits, then nor can $$S$$ have.

Hence $$S$$ is a connected simple graph with no circuits.

Thus by definition, $$S$$ is a tree.