310 CHAPTER 13. THE GENERALIZED RIEMANN INTEGRAL
split again according to whether j ≥ n or j < n. Thus∣∣∣∣S (P, fn)−∫ b
afndF
∣∣∣∣=∣∣∣∣∣ ∞
∑j=N
∑i∈I j
fn (ti)∆Fi−∞
∑j=N
∑i∈I j
∫Ii
fndF
∣∣∣∣∣Next, split further according to whether j ≥ n.∣∣∣∣S (P, fn)−
∫ b
afndF
∣∣∣∣≤∣∣∣∣∣ ∞
∑j=n
∑i∈I j
fn (ti)∆Fi−∞
∑j=n
∑i∈I j
∫Ii
fndF
∣∣∣∣∣ (13.9)
+
∣∣∣∣∣n−1
∑j=N
∑i∈I j
fn (ti)∆Fi−n−1
∑j=N
∑i∈I j
∫Ii
fndF
∣∣∣∣∣ (13.10)
By 13.8,
≤ η +
∣∣∣∣∣n−1
∑j=N
∑i∈I j
∫Ii
fndF−n−1
∑j=N
∑i∈I j
fn (ti)∆Fi
∣∣∣∣∣ (13.11)
Next split up the last term in 13.11.∣∣∣∣∣n−1
∑j=N
∑i∈I j
∫Ii
fndF−n−1
∑j=N
∑i∈I j
fn (ti)∆Fi
∣∣∣∣∣≤
∣∣∣∣∣n−1
∑j=N
∑i∈I j
∫Ii
fndF−n−1
∑j=N
∑i∈I j
∫Ii
f jdF
∣∣∣∣∣ (13.12)
+
∣∣∣∣∣n−1
∑j=N
∑i∈I j
∫Ii
f jdF−n−1
∑j=N
∑i∈I j
f j (ti)∆Fi
∣∣∣∣∣ (13.13)
+
∣∣∣∣∣n−1
∑j=N
∑i∈I j
f j (ti)∆Fi−n−1
∑j=N
∑i∈I j
fn (ti)∆Fi
∣∣∣∣∣ (13.14)
Then 13.12 is∣∣∣∑n−1
j=N ∑i∈I j
∫Ii fndF−∑
n−1j=N ∑i∈I j
∫Ii f jdF
∣∣∣=n−1
∑j=N
∑i=I j
∫Ii( fn− f j)dF ≤
∫ b
afndF−
∫ b
af jdF ≤ I−
∫ b
afNdF < η
Next consider 13.13. This term satisfies∣∣∣∣∣n−1
∑j=N
∑i∈I j
∫Ii
f jdF−n−1
∑j=N
∑i∈I j
f j (ti)∆Fi
∣∣∣∣∣≤∞
∑j=N
∣∣∣∣∣ ∑i∈I j
∫Ii
f jdF− ∑i∈I j
f j (ti)∆Fi
∣∣∣∣∣≤ ∞
∑j=N
η2−( j+1) ≤ η