The brief
The Audi had a check engine light on, and the owner had noticed he was filling up more often than usual, with a rougher idle and a hesitation when he asked for acceleration. He brought it in for a proper diagnosis rather than guessing. That combination, the light plus worse fuel use plus rough running, is the kind of thing a failing oxygen sensor causes. The O2 sensor reads the exhaust and tells the engine computer how rich or lean the burn is, so the computer can trim the fuel to keep it right. When the sensor goes lazy or stops reading properly, the computer gets a bad picture and trims the fuel wrong, which is the rough idle, the hesitation and the extra thirst, and it flags the fault. A sensor doing that has had its run.
The diagnosis
Comprehensive diagnostics pointed at the oxygen sensor: its readings were slow and out of range, not the lively switching a healthy sensor gives, so the engine was running off a bad signal. The rest of the fuel and ignition side checked out, so the sensor itself was the fault, not something it was reacting to. That's a replacement. An O2 sensor is a sealed part with a heating element and a sensing element that wear out, you don't rebuild it, and a lazy one only gets lazier, so it was getting changed.
The work
The old O2 sensor was unthreaded from the exhaust and a new genuine Audi-spec sensor fitted in its place, the connector clipped back home. Then the stored fault codes were cleared and the live data checked on the scanner to confirm the new sensor was switching cleanly and the fuel trims had settled back to normal. A road test followed to confirm the idle was steady and the hesitation was gone.
The outcome
Steady idle, the throttle responding cleanly, fuel economy back to normal across the next tank, and the check engine light out after a drive cycle. The Audi went home running cleanly. A failing O2 sensor quietly costs fuel and roughens the running while the light nags, so swapping it and letting the computer relearn put the running right and cleared the fault for good.