196 CHAPTER 9. INTEGRATION

− f(t ′j+1

)(g(x j)−g(z))+

n

∑i= j+1

(f (ti)− f

(t ′i+1))

(g(xi)−g(xi−1))

The term, f (t j)(g(x j)−g

(x j−1

))can be written as

f (t j)(g(x j)−g

(x j−1

))= f (t j)(g(x j)−g(z))+ f (t j)

(g(z)−g

(x j−1

))and so, the middle terms can be written as

f (t j)(g(x j)−g(z))+ f (t j)(g(z)−g

(x j−1

))− f(t ′j)(

g(z)−g(x j−1

))− f

(t ′j+1

)(g(x j)−g(z))

=(

f (t j)− f(t ′j+1

))(g(x j)−g(z))+

(f (t j)− f

(t ′j))(

g(z)−g(x j−1

))The absolute value of this is dominated by

<ε

2(V[a,b] (g)+1

) (∣∣g(x j)−g(z)∣∣+ ∣∣g(z)−g

(x j−1

)∣∣)This is because the various pairs of values at which f is evaluated are closer than δ . Simi-larly, ∣∣∣∣∣ j−1

∑i=1

(f (ti)− f

(t ′i))

(g(xi)−g(xi−1))

∣∣∣∣∣≤ j−1

∑i=1

∣∣ f (ti)− f(t ′i)∣∣ |g(xi)−g(xi−1)|

≤j−1

∑i=1

ε

2(V[a,b] (g)+1

) |g(xi)−g(xi−1)|

and∣∣∣∣∣ n

∑i= j+1

(f (ti)− f

(t ′i+1))

(g(xi)−g(xi−1))

∣∣∣∣∣≤ n

∑i= j+1

ε

2(V[a,b] (g)+1

) |g(xi)−g(xi−1)| .

Thus renumbering the points to include z,

∣∣S (P, f )−S(P′, f

)∣∣≤ n+1

∑i=1

ε

2(V[a,b] (g)+1

) |g(xi)−g(xi−1)|< ε/2.

Similar reasoning would apply if you added in two new points in the partition or more gen-erally, any finite number of new points. You would just have to consider more exceptionalterms. Therefore, if ||P|| < δ and Q is any partition, then from what was just shown, youcan pick the points on the intervals any way you like and

|S (P, f )−S (P∪Q, f )|< ε/2.

Therefore, if ||P|| , ||Q||< δ ,

|S (P, f )−S (Q, f )| ≤ |S (P, f )−S (P∪Q, f )|+ |S (P∪Q, f )−S (Q, f )|< ε/2+ ε/2 = ε