32 pitcher: interrelations of 



(2) follows from §26.1 ; §23a3i; §31.1. (3) follows from §23011,; § 31.2. 

 (4) is a corollary of (3). We state (3) as 



Theorem I. // a class 90? of functions on ^ has the properties iv;^, 

 if 2* relative to the development A then the ^-extension of SK contains the 

 class of constants and possesses the property A. 



32. Propositions. — The following proposition relative to two func- 

 tions yu, a on ^]? with development A is fundamental and is to be 

 compared with the proposition, " every function continuous on a 

 closed interval is bounded," in the ordinary theory of continuous 

 functions. 



(1) A'^=.M''•^^3.M^'"*^ 



viz., if the development A has the property Cn and ii has the property 

 Kicy then ;i is dominated by the class of constants on ^. 

 Proof. — 



ix^-". = . 113 {e :zi -.3 m, a K^^p^„^ . d . A(m;,, - fip,) ^ eAcr,,,). 



For e = 1 we have 



Since A has the property Cj, there exists a stage TOq ^ nii, of the de- 

 velopment which is complete. Also there exists a finite number of 

 elements j}i such that each element po is directly connected* with 

 some one of the elements pi at the stage mo. Therefore there exists 

 a function constant as to 'ip which dominates ju. Proposition (2) 

 follows at once. 



(2) A^= . 5m^= . 3 . 9[)J^'-'<''-". 



In the following proposition relative to a class ^, with a develop- 

 ment A conditioned as indicated, the system 3)(9}?) = ((S"')) is under- 

 stood to be the system whose existence is implied by the fact that 

 50? has the property A. 



(3) A''^ . ^m) . 9JP«'-««. : 3 :. nio . e : 3 : 3 rn, > Wo ^p.^.^l ^^p'' ^ «• 



/I 



Proof. — By condition (lb) of § 17 on the developmental system and 

 5W^'" we have 



M . e : 3 : 3 m^, 9 (m ^ m,,, . p) . d . ^ Ayu,™, 5;"'' ^ e. 



I, 



Recognizing that this holds in particular if ix is any one of the func- 



* Cf. § u. 



