Consider a desk lamp with a switch. The switch turns the light in the lamp on and off. Thompson is able to turn this switch on or off in any interval of time, and can turn it on or off ad infinitum in a finite period of time. He always switches the lamp to its opposite position (from on to off, or off to on) and always in half the time it took the previous time. The lamp starts switched on, and the timer starts. One minute later Thompson turns the lamp off, thirty seconds later turns it on, fifteen seconds later it's off again, etc.
The sum of this infinite series is two minutes. Will the lamp be on, or off at the end of that two minutes?
Extra: For your own consideration, would the answer be different if the lamp had started off?