According to wikipedia, and many other sources I have read through, the Held-Karp algorithm is based on the idea that "every subpath of a path of minimum distance is itself a minimum distance."
I was hoping someone could explain this concept to me, because that claim doesn't seem true to me at all. If every subpath was a minimum distance, then that would mean that every pair of edges (3 connected cities) would itself be a minimum distance, which would mean that those are simply the two best edges connected to each other, which would also mean that a basic greedy algorithm would output an optimal TSP circuit every time.
So, clearly I am missing something conceptually. Help!