Mathematical philosophy, a study of fate and freedom; lectures for educated laymen, by Cassius J. Keyser.

MORE ABOUT LIMITS 275
we know all the couples constituting it; and you note that
its field F is the same as that of S above. Let V be the
same as before and let k be the same as in the above
paragraph, in which O was found to be an S k-limit of V.
The question is: is O an S' k-limit of F? The answer is
no, as you readily see; for choose a k-neighborhood d of 0,
where d is, say, ~; any F term t' differing by more than
null (zero) from 0, if it be in the chosen neighborhood, is,
as you see, a predecessor of -; and so 1, though it is between such t' and 0, is not in the chosen neighborhood;
accordingly, as said, O is not an S' k-limit of F.
Comparing D2 with D3 you observe that D3 contemplates F's range as a part of the field of a sequence and
that D2 does not; you notice, too, that D3 contains the
same conditions as D2 contains and one other-the
" between " condition (which would indeed be meaningless
in D2 inasmuch as D2 does not regard F's range as included
in the field of a sequence). It follows that if a F have a
limit t under D3, the same F has t for limit under D2.
Is the converse true? It is easy and instructive to show
by an example that it is not. Consider the numbers in
the row
(+-I), ( —I), ( +I), ( —I), (+I), (-I)... *;
which are the same as the numbers in the row
3 9 15 33 63, --. -T6, 32, -84,...
let S be a sequence such that, if aSb, a and b are in the row,
a on the left of b, and that each number in the row is an
S predecessor of every number on its right. S's field F
is the class of the numbers in the row. Let k be the