Infinite rooms and doors (2) > Suppose we have a house (with finitely many rooms) in which every room has an even number of doors. Prove that the number of doors from the house to the outside world is also even. All I could figure out up till now, is that each door connects 2 rooms. Can we prove this without using graph theory? (Graph theory answers will do just fine though). I don't want complete answers ... just hints.

The handshaking lemma will do nicely. If we think of the outside as just another (very large) room, then each room is a vertex, each door is an edge, and apart from the outside room, each vertex has an even number of edges.