49 
From the general principle of convergence (c.f. n°. 4 of my previous 
paper) it follows, that uniform convergence exists then and then 
only, if corresponding to every positive number «€ a positive number 
d independent of G can be found, so that every two elements Z 
and H' of G» satisfy the inequality |H—L’'| <e. 
4. Let it now be assumed, that the uniform convergence mentioned 
in n°. 3 does exist and moreover, that every aggregate H has a limit 
Ly as regards the covering by the parts of V; in consequence of 
this there is a number Ly corresponding to every element of W. 
We now first demonstrate, that the aggregate {LG} of the numbers 
Lg possesses a limit L as regards the covering by the parts of V. 
For this purpose we must prove, that corresponding to every posi- 
tive number ¢ such a JV; can be found, that every two elements 
of Vs satisfy the inequality |Lg— Lg|< #, where Lg and Le are 
the numbers of {Lig} corresponding to these elements. If /# and L’ 
are. elements of G resp. G’, belonging to a same aggregate H, 
we have: 
|\Lg—Le@| < |Le—£| + |£— £'| + |E'— Le). 
On account of the uniform convergence of the aggregates G the 
aggregate H can be chosen thus, that | Lg—H#| and | H’— Le | 
are both < ze. As H possesses a limit, we can choose d thus, 
that every two elements E and H#’ of AH) satisfy the inequality 
| # —H’| < ye. For the numbers Lg and Le corresponding to two 
elements of Vs then | Lg—Lg |< te+te+ie=e holds good. 
5. We next prove, that the aggregate {Ly} of the numbers Ly 
(as regards the covering by the parts of W) hkewise has the limit L. 
We therefore proceed from 
BENE Del PLE, 
in which £ is the common element of the aggregates G and H. 
On account of the uniform convergence mentioned in n°.3 we can 
determine the positive number J in such a way, that | H—Lg|< ye 
holds good if # be an element of Gs, ie. if H corresponds to an 
element of Ws; here « is an arbitrarily chosen positive number. If 
we have now chosen a determinate element of W, (by which also 
H and Ly are determined), we can find the positive number d, in 
such a way, that the inequality | L,—| < te is satisfied, if Z be 
an element of Hs, i.e. corresponds to an element of Vs. Further 
we can determine d, thus, that we get |\La—Ll< 4e, if G corre- 
sponds to an element of Vs. By making the aggregate G to 
oe! 
Proceedings Royal Acad. Amsterdam. Vol. XXII. 
