lamp paradox

A lamp is turned on for 1/2 minute, off for 1/4 minute, on for 1/8 minute, etc. At the end of one minute, the lamp switch will have been moved ℵ_0 times, where ℵ_0 is aleph-0. Will the lamp be on or off? This paradox is actually nonsensical, since it is equivalent to asking if the "last" integer is even or odd.