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

260 MATHEMATICAL PHILOSOPHY
rational number; of all the non-null (non-zero in this
example) amounts of difference between t and the R
terms, one of them, say d, is as small as any of them,
obviously; hence no R term differing from t by more than
zero is in the neighborhood d of t; and so V is limitless.
Query: Are there any variables which, under D2, have
limits? The answer is yes; such variables abound in
endless number and variety. Is the x in (I) of our list of
variables one of them? No; for the difference-kind
appropriate to cardinal numbers is discrete. For a like
reason neither is the variable S, in (4) one of them.
But, as you easily see, the variable x, or -I, in (5) has
a limit, namely, zero (0); so has Sn in (6), the limit being
2; so, too, has S. in (7) if r be numerically less than I;
for the range of n is the class: I, 2, 3,...; the range of
r~
r" is the class: r', r2, r3,...; the range of -r is the
r r2 r3
class: r, -r-, r,.; and the range of S. is the class
I -r I -r I -r
/ - r I r2 \ a I r3 \
a\ï-r I-r/1 aî-r I-r]' \î-r a-r
Consider these ranges. Let d be any given number
greater than 0; as r is less than I, it is plain that there are
terms in the range of rn exceeding O by less than d-terms
that is, in any prescribed neighborhood of 0, however
small; and so, O is the limit of r"; evidently it is also the
rn
limit of -.; without further talk you see that in any
prescribed neighborhood of I-r there are terms of the
range of S~; hence S has r as limit. What if r be