for every 7, so that 
| (Bi )—F(Bx)| Se 
Bi Br: ’ 
whence it follows that 7(3;) tends to a limit f(z) as B; tends to z,. 
Let » denote an arbitrary positive number. For all sufficiently 
small values of |8;—z,| we then have: 
FE) SEN SE: 
2G 
Also | S,(3:) — S, (z.) | <5 |8i—z,| for every n, so that for all 
sufficiently small values of |8;—2,| the relation 
SG) Sule) <5 
holds, where n is arbitrary. 
Now choose a 6; satisfying these two conditions, then from a 
certain m onwards we have: 
SE) FEN CT. 
It follows thence that from this value of m onwards we have 
continually : 
Sz) (Z0)| Sn, 
from which we conclude to the convergence of the series at z, 
The sum there is f(z). 
At the same time it has become evident that 
lim f(Bi) = f(20) 
i=Z, 
2. Provisorily we consider n constant. Then for |z—z,|<4R 
we have: 
Sn —S, 0 
A a (z,) in ne) ein ) ze Wn(2) ’ 
Ze 
Sn Zz —S,(z 
where lim w,(z) — 90. The function nd Jobat ‚) 
220 2—2 
is analytical inside 
0 
© U 
(4), its absolute value being less than me If m now is made to 
increase infinitely, this function tends to a limiting value at the 
points @; and therefore, according to § 1, also at z,. At z, it has 
