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You are in hell for all eternity and the devil gives you two dollar bills every day with increasing serial numbers. He then takes the dollar bill you have with the smallest serial number. At the end of your infinite stay in hell, do you have infinite money or no money?

The intuitive answer for me was that if you look at the partial sums, on the Nth day you'll have $N$ dollars. So clearly after infinity days you should have infinity dollars.

On the other hand, you can also prove that for every dollar you ever received, there's a corresponding day that the devil takes it back, leaving you with nothing in the end.

Which one is the right answer?


marked as duplicate by Asaf Karagila Oct 1 '18 at 13:13

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.


Let's say the serial numbers are just counting. I.e.: $1,2,3...$. in this form we can see that he takes at day $m$ the dollar with the serial number $m$

Now let's say at the end you left with a dollar bill, this dollar bill will have a serial number, $n$, that means you day $n$ didn't passed, that is impossible because you are there to eternity. So you will have no money at the end

  • $\begingroup$ There's no such thing as "the end" of an infinite sequence $\endgroup$ – dbx Oct 1 '18 at 13:48
  • 1
    $\begingroup$ @dbx It often makes perfect sense to talk about the end of an infinite sequence - there is no last stage, but we can make sense in many cases (including this one) of the notion of "after the process is complete." This answer is correct, and the point (re: explaining the apparent paradox) is that the "amount of money" function isn't continuous in the way we would naively want. $\endgroup$ – Noah Schweber Oct 1 '18 at 13:57
  • $\begingroup$ As the answers and extended discussion in the duplicate link indicate, it's not so cut-and-dry as that. In this question, something happens every day, for infinitely many days. There is no "end" of that. The duplicate link is a limiting process, and so it makes sense there. There is no "at the end of your infinite stay", just as there is no largest natural number. At best, the answer above is correct for one specific interpretation of the question, but as asked it's ambiguous. $\endgroup$ – dbx Oct 1 '18 at 14:37
  • $\begingroup$ @dbx no, because the OP said the order which the devil take the coins and the order it put them it then this is the only answer $\endgroup$ – Holo Oct 1 '18 at 14:41
  • $\begingroup$ @dbx yes, this answer is not as good as the one in the link, but what made you say it is cut and dry? $\endgroup$ – Holo Oct 1 '18 at 14:43

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