everyday life - Would a bicycle stay upright if moving on a treadmill and why?

I suspect not, because moving forward (or backwards for that matter) is an important part, but I would like to confirm.

UPDATE: Clearly it's possible http://www.youtube.com/watch?v=NuJvH1W7s1Y Can someone explain why?

Answer

Because you can move the bike from left to right, like when you're riding, to balance it.

Whether you're moving or not doesn't matter, just that your wheels are turning so the steering works.