Math Puzzle

/sci/ has been getting boring recently.  Let me attempt something new.  This is a math puzzle I heard about ages ago.

There are N boxes, each with a real number such that no two boxes have the same number.

The first box is opened so that you can see its contents.  You can a) choose this box or b) discard this box and open the next one [with which you repeat the procedure].  If you choose a), and the current box contains the largest number, then you win $1000.  If you choose b), then you will not be given the option of going back to the discarded box.  Of course, discarding all the boxes means that you lose by default.

What is the strategy to optimize your chances of winning, and what is this probability?

N=1, P=1
N=2, P=1/2
N=3, choosing the 1st box gives P=1/3, but if you discard the 1st box and choose the next largest box, you get P=3/6=1/2 > 1/3.
N>3, ...?

There is no interesting strategy given just this information.  The optimal strategy is just to pick the first box that has a non-zero amount of money.

Just consider N=2.  Say I give you the first box and say it has 10 dollars in it.  According to the parameters of the game the 2nd box could have any amount of money except exactly 10 dollars and we know nothing other than that.  This information does not provide a means to assign a probability distribution to the possible outcomes of opening the 2nd box, i.e. assigning a distribution to the range [0,infinity)\{10}.