Apples and Infinity I am taking a proof writing and discrete mathematics course and we are learning about infinity. My TA asked me the following question and I'm wondering if my solution is correct? 
Question:Suppose Person A has infinitely many apples and Person B has ${0}$ apples. Person A wants to play a game with Person B in which there are infinitely many rounds. In each round Person A gives Person B ${10}$ apples and Person B gives ${1}$ back. What is the least number of apples Person B must have at the end of the game?
My Solution:
Zero is the minimum. There is a way to play in which Person B will have ${0}$ apples: Suppose (to start) all of the apples that Person A has are labeled 1 to ${\infty}$. In round one Person A gives apples labeled ${1-10}$ to Person B, in round two ${21-30}$, and so on... Further, in each round Person B gives back the apple labeled with that round number: i.e. in round one he returns the Apple Labeled ${1}$, etc...
My thought is that Person B will have ${0}$ apples at the end because if anyone disagrees, they'd have to say B has at least one apple. But which apple would B have? The apple labeled ${1000}$? No B handed that back in round one-thousand. The apple labeled ${1000000}$? No, B handed that back in round one million. And so on such that B has no apples.
 A: This is a terrible question. Firstly, it speaks of infinity but seems to mean countable, reinforcing the false idea that infinity is a number and that there is just one infinity. Secondly, it is not a mathematical question but rather a philosophical metaphysical question. Here is a similar situation: suppose a light switch is switched at time intervals $1/2^n$, for each $n$, counting in minutes let's say. Every time the switch is switched, a light bulb changes between on and off. The process will end after precisely $2$ minutes. Is the light on or off after 3 minutes? Does it matter if the light was initially on or off?
This is a metaphysical question, not a mathematical one. If you try to model it mathematically it becomes very clear. Initially the light is either on or off, signified by $1$ or $0$, respectively. Then the entire process is clearly modeled by an alternating sequence of $0$ and $1$. So we have the sequence $(s_n)_{n\ge 1}$ of which we know is an alternating sequence, and we seem to ask what is $s_\infty $. Well, the answer is that this is a nonsensical question. The situation after the process was done was never modeled, so it's not there. Interpreting $s_\infty $ as a limit does not help either. If you try to add the final state of the light into the model you run into the physical problem of actually modeling a manifestly physically impossible scenario. 
