OSN 2009 Problem 4

I proposed this problem to OSN (Indonesian Science Olympiad) 2009 committee and it was selected as problem #4.

There are 7 cities that are connected by a railroad network. Each railroad segment connects two cities, and each city is connected by at least 3 segments. Prove that there is a route that visits exactly 4 cities, visits them exactly once, and goes back to the city of origin.

(Example: A-B-C-D-A)