I am looking for a proof of a particular theorem. The theorem would state:
Every finite, weakly connected digraph contains at least one source or cycle.
Since every finite graph is either cyclic or acyclic, an alternative statement would be:
Every finite, weakly connected, acyclic digraph has a source.
with the previous theorem as a corollary.
Note: Assuming that the above is correct, then for any natural number $n$, an effective procedure exists to show that every weakly connected, acyclic digraph with fewer than $n$ vertices has a source - namely, "pick a point and backtrack". While this makes the theorem seem trivial, it is insufficient to prove that every finite, weakly connected, acyclic digraph has a source, hence the request for a proof.
I checked the library, all of my textbooks, and searched online, but I can't seem to find any reference to this theorem. The notion seems sufficiently obvious so as to be a fundamental result in graph theory, but no one I've spoken to has been able to put a name to it.
Is there a name for this theorem (or other, more significant theorem from which it can be easily derived), and where can I find a proof of it?