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d denoting an arbitrary positive number it follows from the assumption 
c, that a value NV, can be assigned such that for n > N, we have 
| Una (C) + Unpe (Ce) +... + Ur (Ce) | << $d. From the assumption 
6 it follows moreover that a number ae can be found, such that for 
n > N, the inequality | A, (8) | <a on holds good. 
Let N be the greater of the M2 numbers MN, and JN,, then 
| Up (©) + Une (1) + .... + une (2) | << d for every n > N and 
for every « within the interval 7. This establishes a; the assumption 
d can here be dispensed with. 
3. Proof of @. If we put u, (@) + u, (#) + .... =p (e) and 
u, (a) Hw, («) +... =p (e) then the assertion 8 expresses that 
… @(p + h) — p (p) 
If we put u, (x) + u, (©) +... + un (e)—= U,(@) and un (we) + 
+ Ure (w) + ....= On (#) then 
= ee EN 
DET ol) je ae hae 
= w(p + Oh) — w(p)— Ri (p + Oh) + © as a EE 
hence: 
SE Swee) + |Bu(p+Oh)\ + 
+ [ree oak pe 
In consequence of the assumptions 6 and d yw (2) is continuous for 
x =p. Hence, d denoting an arbitrary positive number, the positive 
number ¢ can be assigned such that | y(p + h) —yVp) | <4 for 
hi <<; for | h| <e we have then also | w(p + Gh)—wy(p) | << 40. 
In consequence of the assumption 6 we may assign AV, such 
that for any „>> N, and any @ in the interval 7 the inequality 
| Ane) | <4 is satisfied. For n > N, then | B, (p + OA) | < id. 
Let / be a definite number satisfying | h | <« and h #0, then 
the numbers MN, and MN, can be so determined that for > MN, and 
n > N, the inequalities jo, (p + h)| <4 d|h| resp. | en (p) <4 d|h 
are satisfied. 
Now, if n be chosen larger than the largest of the numbers N,, 
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