Principia mathematica, by Alfred North Whitehead ... and Bertrand Russell.

718

SERIES

[PART V

P-persistent. In this case, x may be regarded as not greater than the
"ultimate " values of the function. Now the class of arguments concerned
is C'Q n C(R. Hence the class of P-persistent terms is
p'P*,'"R'"Q*C"(C('Q n Ia'R),
where the factor C'P may be added in order to accommodate the formula to
the trivial case where C'Q n\ CR = A (the only case in which the factor C'P
makes any difference). Thus the class of P-persistent terms is the limiting
section. Similarly the P-persistent terms are the limiting upper section.
These are the terms which are not less than the "ultimate" values of the
function. Thus the product PRoQ is the terms which are neither greater
than all ultimate values, nor less; hence it is the class of ultimate values,
which may be appropriately called the " ultimate oscillation."
It will be seen that PRoQ, being the product of an upper and lower
section, is itself a stretch: we may call it (alternatively) the "limiting
stretch." It consists of all members x of the P-series such that the function
does not, however great we make the argument, become and remain less
than x, nor yet become and remain greater than x. If PRoSQ consists of a
single term, that term is the limit of the function as the argument travels up
the series Q. (This is, of course, in general different from the limit of the
values of the function considered simply as a class of members of C'P, i.e. it
is different from ltp'R"C'Q.) If PRosQ does not consist of a single term or
none, we shall have two limits to consider, namely limaxp'PRosQ and
liminp'PRo8Q. which give the two boundaries of the ultimate values of the
function. When the class PRosQ is null, the function may be regarded as
having a definite limit: in this case, PRScQ and PRScQ are the two parts of
an "irrational" Dedekind cut, i.e. a cut in which the first portion has no
maximum and the second no minimum. Thus PRoSQ e 0 v 1 is the condition
for a definite limit of the function as the argument grows indefinitely.
The above gives the generalization of the limit of a function when the
argument may be any member of C'Q n ('R. In order to obtain limits for
other classes of arguments, it is only necessary, as a rule, to limit the field of
Q to the class of arguments in question, i.e. to replace Q by Q a (cf. *232).
In order, however, to avoid vexatious and trivial exceptions arising when a e 1,
it is more convenient to replace Q by Q*, a. Thus the section of P defined
by the class of arguments a is PR,,(Q* t a). We put
(PRQ)SC= PR, (Q a) Df.
This definition is useful because we very often wish to be able to exhibit the
limiting section defined by a as a function of a. The section (PRQ),,'a is
such that, if x is any member of it, and y is any argument belonging to a,
there is in a an argument equal to or later than y, for which the function