560 
SUB- AND INFRA-UNIFORM CONVERGENCE 
At a point b of 53, there exists by hypothesis a Vs(b) and a X 0 
such that for each X > X 0 
| F K (x) \ < e , for any x in V s (b). 
Let C btX be a cube lying in L> s (b), having b as center. Since 53 
is complete there exists a finite number of these cubes 
a 
Ml 
O, 
(1 
such that each point of 53 lies within one of them. 
Moreover , — 
\F Xk (x) | < e, 
for any x of Si lying in the /e th cube of 1). 
As B D embraces but a finite number of cubes, and as the same 
is true of 1), there is a finite set of layers £ such that 
| F v (x) | < e , in each 8. 
The convergence is thus subuniform, as X, g are arbitrarily large. 
2. The reasoning of the preceding section gives us also the 
theorem: 
Let 
lim f (x x ••• x m , f • •• tj = 0(ar, ••• x m ) 
in 36, t finite or infinite. Let the convergence be uniform in 36 except 
possibly for the points of a complete discrete set Q? = \e\. For each 
point e, let there exist an g such that setting e (x, 0 =/0> 0 - <£0*0, 
lim e(x, t) = 0 , for any t in F v *(r). 
x=e 
Thenf= (f> subuniformly in 36. 
3. As a special case of 1 we have the theorem : 
Let F(x) =f i {x) +/ 2 <>) + ••• 
converge in 31, and converge uniformly in Si, except at x = a 1 , ••• x = a s . 
At x = a t let there exist a v L such that 
lim F ni (x) — 0 , n t > v L 
x= a L 
Then F converges subuniformly in Si. 
i = 1, 2 s.