47 
sponding to every positive number & a positive number d can be found 
so that every pair of elements E and E’ of Ve satisfies the inequality 
| E—E’| < (GENERAL PRINCIPLE OF CONVERGENCE). That this condition 
is necessary for the existence of a limit is obvious. 
That this condition moreover is sufficient, appears in the case of 
real numbers by noticing that (the condition being fulfilled) a number 
satisfying the definition of a limit is furnished by the upper bounda- 
ry of the numbers a, for which there exists a Vs all elements of 
which are >a. In the case of complex numbers the theorem is 
further proved by applying the theorem for real numbers to the 
real parts and the coefficients of 7 of the complex nnmbers. 
5. let V and W be two aggregates of numbers, the elements of 
which being placed into correspondence. We suppose the coverings 
of the positive numbers by the aggregates of the parts of V resp. W 
to be of such a nature, that for each positive number d the parts 
Vs and JW, in the correspondence between V and W are corre- 
sponding ones. 
_We now form an aggregate U by adding the corresponding ele- 
ments of V and W, at the same time transferring the covering to 
U. If now with these coverings V shows a limit L, and W a limit 
Lo, then also U has a limit, viz. L, + Lp, as may easily may be 
deduced from the definition of limit. 
Other known limit-theorems also can be stated in this manner in 
a general way. 
6. We now suppose, that the elements of the aggregate V are 
real numbers. About the existence however of a limit of V, as regards 
the chosen covering, nothing is assumed to be known. We can then 
consider the lower boundary of the upper boundaries of the aggre- 
gates Vs, and call it the upper Limit of the aggregate V as regards 
the considered covering. The upper limit B is + oo, if all aggregates 
Vs are unbounded to the right, and — oo, if the aggregate of the 
upper boundaries of the aggregates Vs is unbounded to the left. 
We likewise call the wpper boundary O of the lower boundaries 
of the aggregates Vs the Lower Limit of V as regards the considered 
covering; this lower limit can also be + o. It is easily proved, that 
B and O always satisfy the inequality OS B. 
The aggregate V has a limit Z then only, when O and B are 
equal and finite; we then have L = O= BL. If O and B are both 
+o, we speak of an IMPROPER LIMIT + oo, if O and B are both 
— oo, Of an IMPROPER LIMIT — o. 
