Counting - puzzle question Suppose that you have inﬁnitely many one dollar bills (numbered
1, 3, 5, . . . ) and you come upon the Devil, who is willing to pay
two dollars for each of your one-dollar bills.
The Devil is very particular, however, about the
order in which the bills are exchanged. The contract stipulates that
in each sub-transaction he buys from you your lowest-numbered
bill and pays you with higher-numbered bills.
First sub-transaction takes 1/2 hour, then 1/4 hour, 1/8, and so on,
so that after one hour the entire exchange will be complete.
How can this deal be harmful?
 A: At the end of the hour you have none of the numbered bills. Let $n_k$ be the lowest-numbered bill in your possession just before the $(k+1)$-st sub-transaction. Thus, $n_0=1$. The rules imply that $n_k<n_{k+1}$ for each $k\in\Bbb N$. Thus, for any $m\in\Bbb N$ there is a $k\in\Bbb N$ such that $m<n_k$, which means that after sub-transaction $k$ you definitely do not have bill number $m$ in your possession. Since $m$ was arbitrary, you have none of the numbered bills at the end of the hour.
A: I don't agree that it's harmful.  It's true that, as Brian points out, there exists a $k$ for each $m$ where you have no bills numbered $m$ or lower.  But, at the same time, you will have infinite ($\aleph_0$, to be pedantic) bills, regardless.
If we take the limit as $k$ approaches infinity of your smallest bill number, it approaches infinity.  But the cardinality of the number of bills you have remains infinite the whole way through.  (And the economic value is of course a function of the cardinality, not the smallest bill number).  Finally, the set of all bill numbers you have has no limit, so, from this perspective, the question isn't well defined.  (Unless, of course, it's a joke, in which case the jokes on me for writing this up!)
