Mathematics. — “A general dejinition of limit with application 
to limit-theorems”*). By Prof. Frep. Scnun. (Communicated by 
Prof. J. CARDINAAL). 
(Communicated in the meeting of May 3, 1919). 
1. Let us assume an aggregate V of real or complex numbers, 
in which the same number may occur repeatedly. This can take 
place, if a mode of arising of the numbers of V has been given 
and various modes of arising may lead to the same number. Those 
equal numbers however, are considered as different elements of JV, 
so that we distinguish between a number having arisen in the former 
and the same number having arisen in the latter manner. 
2. Next we assume a law given by which every positive number 
JS is made to correspond to a part Vs (consisting of at least one 
element) of V (covering of the aggregate of the positive numbers 
by the aggregate of the parts of VV), in this way, that Vw isa part 
of Vs, if d' <d; here Vs is called a part of V if each element 
of Vs is an element of V, so that the part can be identical with 
the whole aggegrate. If now L is a (real or complex) number with 
the property that corresponding to every positive number d there exists 
such a posite number e that every element KH of Vo satisfies the 
inequality |H—L| <e«, then L is called the ramir of the aggregate 
V as regards the covering by which Vs is made to correspond to 0. 
It is clear that there can be at most but one number with the before- 
mentioned property. 
3. The covering observed in n°. 2 we call EQUIVALENT to a second 
covering by which a positive number d is made to correspond to a 
part V's of V, if corresponding to every positive number d a positive 
number Jd, can be found so that V's, ts a part of Ve and Va, ts 
a part of V's. It is evident that this equivalence is a transitive one. 
It furthermore easily appears that two equivalent coverings both gwe 
the same limit, or no limit exists in the two cases. Hence equivalent 
coverings can be considered as the same limittransition, so that with 
& MANNER OF LIMITTRANSITION we mean a set of mutually equivalent 
coverings of the kind mentioned in n°. 2. 
4. The limit defined in n°. 2 exists, and exists only then, if corre- 
1) For further particulars c.f. Hand. van het Nat. en Geneeskundig Congres 
1919. 
