 (You might think that this problem could be fixed by taking the elements of the chains to be segments with no endpoint to the right. Then the first of the two chains we considered no longer has the half-way point in any of its segments, and so does not pick out that point. The problem now is that it fails to pick out any part of the line: the previous reasoning showed that it doesn’t pick out any point greater than or less than the half-way point, and now it doesn’t pick out that point either!)