The totality of real numbers is a field (Kemper), i.e., a set of elements upon which can be performed the operations of addition and multiplication, according to the well-known rules.' We suppose also known the derived operations of subtraction, division and raising to a power. A special symbol is used to dis tinguish the numerical from the algebraic value of a number: if x is _positive, VI°x, if x is negative, — ix] =x. The relation of greater and less is indicated by the symbols <, > ; alb means a is greater than b, a
Principle of Continuity.— If [x] is any set of real numbers all less than a certain number, b, then there is a number, x, such that every number of [x] is less than or equal to x, and such that x is the smallest number of which this is true. In other words, x=x < — for every x of[x], and if
there is a number x of [x] such that e
Any number, such as b in the statement above, which is larger than every number of a set [x] is called an upper bound of [x] ; a num ber smaller than every number of a set [x] is called a lower bound of [x]. In this language the principle of continuity says that every set having any upper bound has a least upper bound. It follows immediately that every set having a lower bound has a greatest lower bound. The greatest lower bound or least upper bound of a set .may or may not belong to the set. Thus, zero is the greatest lower bound of the set of all positive numbers but does not belong to it: on the other hand, zero is the least upper bound and does belong to the set of all real numbers which are not positive.
A very large proportion of the advances in critical sharpness of the modern analysts over those of the 17th and 18th centuries is due to the recognition of places where the earlier analysts had unconsciously assumed, or had neglected to apply, the principle of continuity. The form in which this postulate is here stated is due to K. Weierstrass Other forms have been given by R. Dedekind' and G. Can tor•. Indeed, the theory of continuity and irra tional numbers is due about equally to these three great Germans.
Segment, Interval, Limit Point— By the method of analytic geometry (q.v.) there is a correspondence between the totality of real numbers and the points of a line; likewise be tween the set of all pairs of numbers (x, y) and the points of a plane, the pair (x, y) cor responding to the point of which x and y are the co-ordinates. On account of this corre
spondence it is customary to use geometrical language in the theory of real functions; thus, the word point is used interchangeably with the word number. A linear segment is the set of all numbers x, a < x < b or the set of all points between a and b; a planar segment is the interior of a parallelogram with sides parallel to the co-ordinate axes, or the set of all num ber-pairs (x, y), a <
Among the most important consequences of the principle of continuity is the following theo rem which was first stated in a slightly different form by E. Borel 11 in 1895. Since its method is essentially involved in the proof of the theo rem of uniform continuity (see below) given by E. Heine' in 1871, it is usually called (after A. Schoenflies') the Heine-Borel theorem: ((Every set of segments [a] such that every point of an interval,
is an interior point of at least one a, has a finite subset el,
• - ••
such that every point of i is an interior point of at least one
(k== 1, 2, . . . , is).)) We shall prove this theorem only for the linear case, the proof of the planar case being quite analogous. Let the end points of the interval i be a and b, a4b. Let [bl be a set including every point, b of i such that the interval ab' is contained in a finite number of segments of [ci]. There is at least one b'; for one of the end points of any segment of[e] which includes a is evidently a point of jb']. By the principle of continuity, the set of all points, [11], has a least upper bound B, B b. We shall prove that B is a point of [b'] and that
Let a" b", a"