He pursued them through a variety of simulated envornments on the holodeck, such as the streets of Bajor, and up a mountain. That's when Rutherford and Tendi realized that the conditions of the holoprograms were actually having an effect on him physically, as he was tired from climbing the stairs of the mountain.
This gave Rutherford an idea, and he started a new program, one that was a very Arctic-like environment, which caused him to slow down due to extremely low temperatures. Rutherford then overpowered Badgey, and broke his neck.
After resetting Badgey's program, Rutherford simply assumed that he would have no memory of what happened before, but Badgey kept it a secret, as he still remembered Rutherford broke his neck, and he began scheming his revenge. (LD: "Terminal Provocations")
When the Cerritos was attacked by a modified Pakled cruiser, Rutherford once again sought Badgey, this time to code a computer virus to take down the attacking ship. Badgey gave him the virus (after the safety protocols were once again disabled), and Rutherford delivered it in person with the help of Lieutenant Shaxs. While Rutherford uploaded the virus, the process was stalled by Badgey, who said the virus would not upload until Rutherford was killed. Badgey initiated the self-destruct sequence of the ship, but in the final moments, Shaxs ripped off Rutherford's cybernetic implant, and saved the ensign's life, at the cost of his own. (LD: "No Small Parts")