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

MORE ABOUT LIMITS 279
of Fi, of F2 and of F3. It is customary to say, " As Fi
runs along the sequence (ri) from left to right it approaches
I as its limit " or to use some equivalent equally figurative
speech; and similarly for F2 and F3. It is noticed that
the F's in running along their so-called sequences get
nearer and nearer to their limits but never reach them.
The question arises: Is it possible for a sequence having a
limit to be such that the variable, in the course of its
approaching the limit, reaches it one or more times?
Some say no; others say yes. The latter attempt to justify
their answer substantially as follows: Consider, they will
say, the sequence
(r) -, 2, I, î,,, I,... (ad infinitum); I,
got from (r3) by inserting I after each of the successive
pairs of numbers in (r3); observe, they will say, that if a F
runs along (r), skipping the third term, the sixth and so
on, it will approach the same limit (namely, I) as if it
ran along (r3), and that, if it runs along (r) without
skipping, it will again evidently approach the same limit,
I, but in this case will actually reach I infinitely often in
endlessly approaching it; and so you are expected to see
not only that (r) is a sequence having a limit but that,
while endlessly approaching it, it actually reaches it again
and again and again. You instinctively feel that you are
being hocus-pocused by such argument, and your instinct
is sound. What is the trick? It is easy to detect. The
juggler (we may call him a juggler, though he does not
intend to deceive) asks us to regard (r) as a sequence or
at all events as indicating a sequence. Let us try to do
so in good faith. If (r) be or indicate a sequence S, what is
the field F? The answer is obvious: the terms of F are
the numbers in (rs). Among these is I, the final number