My understanding is that the moon is the lesser light (it is an independent light source to the sun, not merely a reflector of sunlight) - so as Thork stated, when the sun can be seen, it is by definition day.
When the sun or the moon move away from an observer on the stationary earth, eventually, the horizontal distance is so great that the angle to the observer tends to 0 and the sun/moon will converge to the horizon. As the atmosphere is not 100% transparent, there will come a distance beyond which light from the sun/moon will no longer reach the observer.
The moon does not in any way depend on the sun to give its light, so can be seen during day and night periods, so long as it is not so far from the observer as to have converged with the horizon.