"427 
Since lim f*O, 2) = f'(O) and lim Fn, (0, =f (O), it is possible to 
z=0 z=0 p 
determine a number J such that 
10, 2) 0, ey le 8 for rd. 
Since at the points of ihe annular region fe 2) <6 the function 
Jr (0, 2) converges to the continuous function f*(O, z), this convergence 
is quasi-uniform there 7. e.: from a finite number of indices > N at 
every point z of the annular region an index n, may be chosen 
which satisfies 
ASO 2) — ft (2) | <e. 
At every point 0 < |z| <5 therefore from a finite number of indices 
>> N such a choice can be made, which establishes the theorem. 
13. Some of the results here obtained we shall apply to Montrr’s 
example, which was cited in the introduction and is reproduced below : 
The function y (2) =n’ze—" tends to zero as n — op for all real values 
1 1 
of z. Now consider the three rectangles J, (— nSearSn,— on <y San i 
n 
1 1 
/ (- nsacn, — <y<n) and LLL, (ns LN nye) 
n 
n 
There exists a polynomium P,(z) which within /, differs less 
1 1 
than 7 from ¢,(z) and within //, and ///, less than — from zero. 
n n 
Evidently the function P,(z) tends to zero for n—>o throughout 
the whole plane. According to the theorem established in § 5 the 
function wy, (a,y,z) must converge quasi-uniformly at the points of 
every set V: P, (#,,y,,0), P,(#,,y,,0),... where a, F0, yv, #0, 
lim «x, =O and lin y,=0. That this is really the case may be pro- 
k=o k= 
ved as follows: Choose an arbitrary pair of numbers ¢, N. Let 
1 
N,be > N and at the same time Br From a certain index &, 
€ 
onwards #, and y, are both internal to /, so that 
1Vo 
23 2 
3 Wy, (es pe Oh Ni (am xy, a Me Me) Ee ; <5 
Hence a number 0 exists such that for | «#,|< 5 ef al | <0 we 
have 
ws Gy ty DIS 
This, we remind, is the = from a certain index %, onwards. 
4M 
Besides, a number NV, > N can be found satisfying N, > — 
me’ 
