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

MORE ABOUT LIMITS ~73
the field F of a sequence S, and (as before) let k be an
available difference-kind; if there be in F a term t such
that, however small a k-neighborhood of t be chosen,
there is in the neighborhood an R term t' differing by
more than null from t and being such that all R terms
between t and t' are in the neighborhood, then t is an S
k-limit of V.
The meaning of D3, which is a bit subtle and sly, may
be made evident by means of a few examples. In adducing examples it will be convenient to make some use
of the customary row method of representing sequences.
For a simple example, let S be the sequence determined
by the propositional function: x is a fraction (having I
for numerator and a positive integer for denominator)
greater than a fraction y (having I or zero for numerator
and a positive integer for denominator). F is composed
of the numbers in the row
4, 1, 4, * * * (ad infinitum), O.
Let V be the variable whose range R is the class of all the
F terms except 0; let k be the kind of difference in respect
of which we compare the values or magnitudes of rational
fractions (as when we say -3 = ). The question
is: Has Y an S k-limit t? The answer is yes: t is
such a limit if t be zero (0). To prove it, suppose chosen a
k-neighborhood d of 0, however small; there is no restriction upon the choice of d save that d must be a positive
rational number; it is plain that there is in R a number t
differing'from O by more than null (zero)and by less than
d; it is evident that such a t and all the R terms between
t and O are in the chosen neighborhood; and hence O
is, as said, an S k-limit of V.
It should be said in passing that a V having, under