I think there were some delivery negligence and errors that caused our newborn injuries. How do we prove it and file a lawsuit?

According to the Arizona statutes, the definition of a "medical malpractice action" is an action for injury against a licensed healthcare provider based on the  provider's alleged negligence, misconduct, errors or omissions. Your brother (through his lawyer) has the burden of proving that:

• The doctor had a duty of care (a standard of care provided by the average Arizona doctor in the same or similar circumstances)
• The doctor failed to adhere to the established standard of care
• The baby suffered a birth injury
• But for the doctor's failure to adhere to the standard of care, the baby would not have been injured (doctor's acts/omissions were the proximate cause of the baby's injuries).

In other words, it is more likely than not that the birth injury occurred because of the doctor's acts or omissions during the birth of the baby.

There are no caps on damages in Arizona. Damages include compensation for pain and suffering, permanent physical disability, and emotional distress (parents are entitled to this damage as well). Furthermore, Arizona courts award the payment of interest to the prevailing party from the date the judgment is entered until the date the judgment is paid.