4.1. UPPER AND LOWER RIEMANN STIELTJES SUMS 45

and the term which corresponds to the interval [xk,xk+1] in U ( f ,Q) is

sup{ f (x) : x ∈ [xk,y]}(F (y)−F (xk))

+sup{ f (x) : x ∈ [y,xk+1]}(F (xk+1)−F (y))

≡M1 (F (y)−F (xk))+M2 (F (xk+1)−F (y))

(4.1.2)

All the other terms in the two sums coincide. Now

sup{ f (x) : x ∈ [xk,xk+1]} ≥max(M1,M2)

and so the expression in 4.1.2 is no larger than

sup{ f (x) : x ∈ [xk,xk+1]}(F (xk+1)−F (y))

+sup{ f (x) : x ∈ [xk,xk+1]}(F (y)−F (xk))

= sup{ f (x) : x ∈ [xk,xk+1]}(F (xk+1)−F (xk)) ,

the term corresponding to the interval, [xk,xk+1] and U ( f ,P) . This proves the first part ofthe lemma pertaining to upper sums because if Q⊇ P, one can obtain Q from P by addingin one point at a time and each time a point is added, the corresponding upper sum eithergets smaller or stays the same. The second part about lower sums is similar and is left asan exercise.

Lemma 4.1.3 If P and Q are two partitions, then

L( f ,P)≤U ( f ,Q) .

Proof: By Lemma 4.1.2,

L( f ,P)≤ L( f ,P∪Q)≤U ( f ,P∪Q)≤U ( f ,Q) .

Definition 4.1.4I ≡ inf{U ( f ,Q) where Q is a partition}I ≡ sup{L( f ,P) where P is a partition}.

Note that I and I are well defined real numbers.

Theorem 4.1.5 I ≤ I.

Proof: From Lemma 4.1.3,

I = sup{L( f ,P) where P is a partition} ≤U ( f ,Q)

because U ( f ,Q) is an upper bound to the set of all lower sums and so it is no smaller thanthe least upper bound. Therefore, since Q is arbitrary,

I = sup{L( f ,P) where P is a partition}≤ inf{U ( f ,Q) where Q is a partition} ≡ I

where the inequality holds because it was just shown that I is a lower bound to the set ofall upper sums and so it is no larger than the greatest lower bound of this set. This provesthe theorem.