176 
SERIES OF FUNCTIONS 
Suppose now | yfr(x, £') | < | yjr(x, t) | for any t' in the rectangu 
lar cell one of whose vertices is t and whose center is t. We say 
then that the convergence of / to is steady or monotone at x. 
If for each x in 21, there exists a rectangular cell such that the 
above inequality holds, we say the convergence is monotone or 
steady in 2i. 
The modification in this definition for the case that r is an ideal 
point is obvious. See I, 814, 815. 
2. We may now state Dim s theorem. 
Let fC x i ••• x m , / ••• t n ) = <£(aq ••• x m ) steadily in the limited com 
plete field 2\ as t = t; t finite or ideal. Let f and be continuous 
functions of x in 21. Then f converges uniformly to </> in 2b 
For let a; be a given point in 21, and 
/O’ i) = fi(x) + y]c(x, £). 
We may take t! so near t that | yjr(x, O I < §' 
o 
Let x' be a point in V v (x). Then 
f(x\ £') = + -fi(x', t'). 
As /is continuous in x, 
Similarly, 
I/O', £') —/O’ OI < !* 
! <ÊO') - <t> O) I < |- 
Thus 
Hence 
| yfr O', O | < e x' in V v (x). 
| tJc (xt) | < e for any x' in V v (x) 
and for any t in the rectangular cell determined by t'. 
As corollaries we have : 
3. Let 6r = 2 |/ H ...t s Oi x m) I converge in the limited complete 
domain 2b Let Gr and each f be continuous in 21- Then & and 
a fortiori F= 2/ tl ... t# converge uniformly in 21, furthermore f H ... t< = 0 
uniformly in 2b 
4. Let Gr = 2 |/ ti ...^Oj ••• xf) | converge in the limited complete 
domain 21, having a as limiting point. Let 6r and each f be con