GENERAL THEORY 
179 
In the figure, the graph of F(x) is drawn heavy. On either 
side of it are drawn the curves F — e, F + e giving the shaded 
band which we call the e-band. 
From 2), 3) we see that the graph of 
each F n , n>m lies in the e-band. The 
figure thus shows at once that 
f 
Fdx 
and 
f 
F n dx 
can differ at most by the area of the 
e-band, i.e. by at most 
2 edx = 2 e(b — a) 
converge 
(i 
sufficient 
have 
(2 
(3 
152. 1. Let us consider a case where the convergence is not 
uniform, as 
o- 
Here •*’.<*)-|5- 
If we plot the curves y = F n (x), we observe that they flatten 
out more and more as n = oo, and approach the rr-axis except 
pleasure for m sufficiently large, 
termwise. But this area is here 
near the origin, where 
they have peaks which 
increase indefinitely in 
height. The curves 
F n (x), n > m, and m suf 
ficiently large, lie within 
an e-band about their 
limit F(F) in any inter 
val which does not in 
clude the origin. 
If the area of the 
— region under the peaks 
could be made small at 
we could obviously integrate