Yes we are the landlords and no no mention of an actual injury or accident.Sounds like it's just a letter (that you received by mail or Fedex or other similar service). Correct? Or were you actually served with a summons and complaint? In other words, have you actually been sued, or has someone merely told you that someone intends to sue you? If it's just a letter, does it provide any detail, or does it just vaguely refer to an "injury as a result of an accident because of negligence."
It seems like a reasonable inference from what you've written here that you're a landlord and that one of your tenants is suing you. Is that correct?
I don't know what "type of proof" might mean, but "proof" isn't needed to file a lawsuit. I can't speak specifically about New York, but the general standard is that a lawyer should not file a lawsuit without evidence of all elements of all alleged causes of action or a reasonable belief that such evidence can be obtained through investigation and discovery.
Sure, but the failure to notify may or may not be legally relevant.
Assuming you have a good landlord's insurance package (or whatever insurance you have), you need to send what you've received to your insurer ASAP and ask that the insurer assign an attorney to represent you.