526 
Mr Young, On Monotone Sequences 
Similarly, by case ( 1 ) 
- - = Lt vf, 
u n = oo 
where v n ' is a bounded positive continuous function, and the 
sequence is monotone decreasing. Thus 
— v = Lt —7 = Lt v n ", 
n = 00 V n n = oo 
where v,/' is a bounded positive continuous function, and the 
sequence is monotone increasing. Thus finally 
v = Lt — v n " = Lt v n , 
n — co iz = 00 
where is a bounded negative continuous f unction and the sequence 
is monotone decreasing. 
Since u n is continuous and never negative and v n bounded and 
continuous, their sum 
Un + V n = f a , say, 
is continuous and has a finite lower bound, so that, though it may 
be + oo , it is nowhere — co . 
Also, since 
U n (&’) ^ U n - |-i (#'), 
and v n (x) ^ v n+1 (x), 
it follows that 
fn if) — U n (x) 4 V n (x) ^ Un+\ (f) 4" 'Vji+i if) ^ fn+i (f)> 
so that the sequence f n ( x ) is monotone decreasing. Also 
Lt f n (^) = Lt [u n (x) + v n (a-)] 
n — 00 71=0 o 
= Lt u n (x) + Lt v n ( x ) 
n= co n = 00 
(since these limits are nowhere both infinite), which 
= u (, x ) + v ( x ) — f(x). 
Thus we have expressed f(x ) as the limit of a monotone 
decreasing sequence of continuous functions f n (x), each of which 
has a finite lower bound. 
Case 4. When f(x) is any lower semi-continuous function, 
— f{x) is an upper semi-continuous function, and can be expressed 
as above, whence the required expression for f(x) follows im- 
mediately. 
This proves the whole theorem. 
Cor. Any upper ( lotver ) semi-continuous function ivhose upper 
(loiver) bound is finite, may be expressed as the limit of a monotone 
descending ( ascending ) sequence of bounded continuous f unctions. 
