524 
Mr Young, On Monotone Sequences 
joining two lower points on the same ordinate, and the same is 
true if one of the second pair of points coincides with one of the 
first. Thus the sequence of continuous functions obtained is mono- 
tone. 
We have only to shew that the sequence has fix) as limit. 
Now by the definition of Baire’s upper limiting function <j> B , 
the construction gives at any primary point 
Bt f n (x) = (f)jj (x), 
n=cc 
and, since f is upper semi-continuous, 
f(x) = <f>B (x), 
which proves that at each primary point, 
Lt f n (x)=f(x). 
n — co 
The same holds at the extremes, where, of course, we are 
concerned with one side only of the point. 
On the other hand at a secondary point x, x n ' and x n " being 
the nearest points of division at the nt h stage, on the left and 
right of x respectively, it follows from the construction that f n (x) 
lies between f n (x n ') and f n (x n "). 
Now f n (xn) was defined as the upper bound of f{x) in an 
interval which clearly contains x as internal point, and, as n 
increases indefinitely, these intervals lie one inside the other and 
have x as sole common internal point. Thus, by the definition of 
Baire’s upper limiting function, 
Bt f n (xf) = (f) B (x). 
71= co 
Since the same is true of Lt f n (x n "), it follows that the same 
71= oo 
is true of Lt f n (x), which is intermediate between the two limits. 
71= co 
Since /is upper semi-continuous, it follows, therefore, that 
Lt f n {x) =f(x). 
71= GO 
This last equation has therefore been shewn to hold at every 
point, which proves that f(x) is the limit of the monotone sequence 
of continuous functions f n (x) constructed. 
This construction was such that f n (x) was never negative, 
since f(x) was never negative, and the sequence was, of course, 
monotone decreasing. If now we replace f n ( x ) b y 
fn O) + ~ , 
the new sequence will still be monotone decreasing and have the 
same limit as before, the generating function is however always 
