After switching a lamp on and off infinitely many times in one minute, is it on or off? So we have a lamp. It's switched on. let's represent its state of being switched on with associating it with $1$ and being off with $-1$. after half a minute passes, you turn it off, after another quarter of a minute passes you turn it on and so on.

Now this process will take : $1/2+1/4+1/16+...=1$ minute

and this is the states of the lamp put into a sequence:

$(1,-1,1,-1,1,...)$

After a minute of switching it on an off passes, will it be on or off?

I'm well aware that the limit of this sequence does not exist, however, my intuition dictates that after a minute passes, it is either on or off. 

So does this lamp have a final state(on or off) which is impossible for us to know? or it does not have a finals state(it seems to me, it must have one!)?
The apparent paradox in this question arises from the conflict between the following two ideas:


Our intuition about physical lamps.  We expect them to have certain properties, such as being on or off, but not both or neither.
A complicated function that oscillates between two values at an increasing pace, whose oscillations become arbitrarily rapid.


It is not possible to have a physical lamp whose on/off state follows the complicated function.  This is the resolution to the paradox.  Thus we must either give up (1) or (2).  If we give up (1), then there is no reason to believe that the state must be either on or off.  The mathematical function has no limit.
Assuming the wiring is 100% efficient and the speed of the electrical current is instantaneous (which is just as illogical as the idea of unreal numbers such as infinite), there would be no final flip, as the time it takes to switch it is always divided by half.

However, in a real-world application, where wiring has latency and electrical current has a set speed, when the time between switches gets low enough that the switch is being turned on then off before any current could pass the wires (assuming any force could act that fast), the overall state will always be off.