. Each of these numbers is "greater" than any of its predecessors, in the following sense:—
One serial number is said to be "greater" than another if any series having the first number contains a part having the second number, but no series having the second number contains a part having the first number.
If we compare the two series
we see that the first is similar to the part of the second which omits the last term, namely, the number 2, but the second is not similar to any part of the first. (This is obvious, but is easily demonstrated.) Thus the second series has a greater serial number than the first, according to the definition—i.e.
is greater than
. But if we add a term at the beginning of a progression instead of the end, we still have a progression. Thus