Advanced Counting Puzzle

Question

Suppose we have a house 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.

Answer 1

Hint: count the number of sides of doors.

Answer 2

It might not be true if we allows doors from a room to itself:

This is equivalent to having a loop in the graph. It also can break down with infinite houses.

(NB. I can't see this image where I am, so hopefully it shows what I intend it to show.)