655 
Let a sequence of functions be given, S,(x),S,(@),...., all 
continuous in a<a<b. Let the sequence converge ina<u<b 
to a function f(x) which is continuous in this interval, and let M 
denote the maximum of f(x) at a, m the minimum. Now, if it is 
possible to conjugate to every number k between m and M a partial 
sequence of the given sequence which at a converges to k, then the 
convergence of the given sequence is quasi-uniform in a<u<b. 
In order to prove this we give ¢, N and on the line =a we 
choose the points P,, P,,... P, such that 
ET Ee hg MEE 
and at the same time : 
P; Pai Je OE I. 
It follows from the supposition that an index n; > N exists for 
which Sn, (a) lies between ZP; and P44. Thus we find » indices. 
Since the functions S,,(x), S,,(z)....S,,_,(2) are continuous at a,a 
number d, can be found such that for z—a < d,, On; lies between 
P; and P14, where 7=0,1,....p—1. Also a number d, can be 
found such that for «—a< d, f(w) lies between P, and P,, for, 
M and m are the maximum and the minimum of f(x) at a. Let 
d<d, and d<d,. For e—a< J, f(x) belongs to one of the intervals 
P; P41. Hence it is possible for every a of the interval a {ew <a + d 
to choose from the wv obtained indices an index mz, such that 
| F(@)—Sn,() ise 
In the same way it is possible to make fora+d<w<ba 
similar choice from a finite number of indices > N, since the given 
sequence in consequence of the continuity of f(«) converges quasi- 
uniformly in this interval. Hereby the theorem is established. 
It is evident that in this theorem the words “to every number k 
between M and N” may be replaced by: “to every number k of a 
set which is everywhere dense in the interval m, M”. 
