10:42 AM PT -- TMZ Sports has obtained the arrest video from the prostitution bust ... and it's clear as day Gainey offered $60 for sex acts to an undercover officer posing as a prostitute.
During the exchange, captured by a hidden surveillance camera, Gainey negotiates a price for a "quick visit" and then tells the officer he's in town "the rest of the week" -- clearly implying he could be a repeat customer.
According to the police report, obtained by TMZ Sports, after Gainey handed over the cash, cops busted in and arrested the golfer .