Volume 1 / Part 5 — *231. LIMITING SECTIONS AND ULTIMATE OSCILLATION OF A FUNCTION.
Reading edition
*231. LIMITING SECTIONS AND ULTIMATE OSCILLATION OF A FUNCTION.
Summary of *231.
In the present number we are concerned with the limiting section defined in a series , to which the values of a function belong, as the arguments to the function increase in the argument-series . That is, we are concerned with the section consisting of those terms of which are such that, however great the argument to becomes, there are still values at least as great as . Such terms as may be said to be -persistent; is -persistent if the function does not ultimately become and remain less than . The class of persistent terms is called the limiting section. The limiting section may be defined as follows. If is any class into which is -convergent, then the section is such that the values of the function are ultimately contained in it. The product of such terms as is the smallest section having this property. Hence if be any member of this section, then ultimately (i.e. for arguments far enough along the series) the values of the function do not persistently remain less than in the series. Thus the product of such terms as is the limiting section, and we may therefore put where the letters "" are intended to suggest "section." (The factor on the right is superfluous except when , i.e. when .)
We will call the limiting section of , i.e. , the "limiting upper section." It will be seen that if is a member of , then the function does not ultimately become and remain, as far as some of its arguments are concerned, greater than , that is, however great we make the argument, we still find values not greater than . Hence if belongs to both and , we find values not less than and values not greater than however great we make the argument. This class, , may therefore be regarded as the class of ultimate values of the function. We will call it the "ultimate oscillation" of the function, since, as the argument approaches , the value of the function ultimately oscillates in this stretch of , and no smaller stretch has the same property. We will denote this class by "," where "" is intended to suggest "oscillation." is a stretch in , because it is the product of two sections. Hence we shall also call it the "limiting stretch." When the function has a definite limit as the argument approaches , the limiting stretch must not contain more than one term.
Limits of functions for arguments in the middle of , which will be considered later, are derived from the limits considered in the present number by limiting the field of to predecessors of .
In this number we prove the following propositions among others:
*231·103.
*231·12.
*231·13.
*231·141.
*231·191.
*231·192.
*231·193.
This proposition is frequently used in the present section.
In all ordinary circumstances, we shall have , so that if the upper and lower limiting sections do not have more than one term in common (i.e. if , they define a Dedekind cut in . The following propositions are concerned with this fact:
*231·202.
*231·21.
*231·22.
Note that "" is the hypothesis that for arguments belonging to , the values belong to .
*231·24.
*231·01.
*231·02.
*231·1.
*231·101.
*231·102.
*231·103.
*231·11.
*231·111.
*231·112.
Dem.
*231·113.
If is a one-valued function (i.e. a one-many relation), and if we write for , and for , we have That is, belongs to if, for any argument in , we can find an argument greater than or equal to , for which the value is greater than or equal to .
*231·12.
This is usually the most convenient formula for .
*231·121.
Dem.
*231·13.
*231·131.
*231·132.
Dem.
*231·133.
*231·134.
*231·14.
*231·141.
Dem.
*231·142.
Dem.
*231·143.
*231·144.
*231·15.
Dem.
*231·151.
*231·152.
The hypothesis is verified not only when , but also under certain more general hypotheses. Two such hypotheses, namely are considered in the following propositions.
*231·153.
Dem.
*231·154.
Dem.
*231·155.
Dem.
*231·156.
Dem.
*231·16.
*231·161.
*231·17.
Dem.
*231·171.
*231·18.
Dem.
*231·181.
Dem.
*231·182.
Dem.
*231·19.
Dem.
*231·191.
*231·192.
*231·193.
This proposition is of fundamental importance.
*231·2.
Dem.
This proposition is fundamental in the theory of limiting segments.
*231·201.
*231·202.
Dem.
*231·21.
Dem.
*231·22.
*231·23.
Dem.
*231·24.
Dem.
*231·25.
*231·251.
*231·252.
*231·4.
Dem.
*231·41.
Dem.