I don't get that at all... Yeah, giving someone a shove is the most minor case of assault imaginable, and it's understandable for the police to dismiss it...What sort of action are you looking for? While inexcusable behavior on the pusher's part, this seems like a pretty minor case of assault (about as minor as it could get) resulting in a pretty minor injury (a single, small bandage and likely some bruising to boot, hence the limp). Even in the ultra-tort-mania US, I have a hard time imagining the police doing more than filing a report for a situation like this unless there are significant medical bills involved.
But pushing someone down a staircase can result in them being paralysed or killed. Pushing someone down a staircase is almost attempted murder.
If someone deliberately tried to kill me, I think I would likely lose my mind. If the police wouldn't act, and if there was no way to remove this person from your life, then you'd be forced to take matters into your own hands...
If this was at work, you'd immediately have to go to the highest authority you can and report to them that a co-worker has been bulling you and it's climaxed with a potentially deadly assault. If they then failed to act, well, you could either pack up and leave, or put a billiard ball in a sock and wait for this person to go to the bathroom...
If it wasn't deliberate, you could go and confront them and explain that they could've easily killed you with that stunt, and if it ever happens again, then you'll <insert ultraviolence here> to them
I had a supposed-friend once pull a stupid stunt that they knew could've very easily got me killed, but they did anyway just for fun, revealing their true personality. Needless to say after that I simply cut them out of my life and never contacted them again... However, my instant reaction at the time was to give them an open-hand slap, resulting in a sting for a few minutes (no wounding). They reported this "assault" to the police and I was forced to go and apologize in front of a magistrate for it (without a chance to explain myself), pay a fine, and attend anger management classes -- under an intimidating police interrogation, I was offered that "bargain" and threatened with worse if I tried to defend myself in court.
Another time when I was younger, I got mugged by a gang of 5 while walking through the city late at night. I had no money, and snapped my bank card in half when they started dragging me to an ATM... So they punched me down and stomped on my head (again, something potentially deadly). However, when I reported this to the police, along with full names and descriptions of the muggers, they told me it wouldn't be worth them investigating and I should just take this as a lesson not to walk dark streets at night...
So -- given my two experiences with the police and justice, I don't have much faith in them either