
“real: chapter_08” — 2011/5/22 — 23:34 — page 55 — #55
Riemann Integration 8-55
Hence |S(P, f , α) −
b
a
fdα| <whenever d(P)<δ. This shows that
lim
d(P)→0
S(P, f , α) =
b
a
fdα.
We now prove the converse with P ={x
0
, x
1
, ..., x
n
} (a partition of
[a, b]) given >0 we can choose s
i
, t
i
∈ (x
i−1
, x
i
) with
f (s
i
)<m
i
+
α(a) − α(b)
and f (t
i
)>M
i
−
α(a) − α(b)
.
If S
1
(P, f , α) =
n
i=1
f (s
i
)(α(x
i
) − α(x
i−1
)) and S
2
(P, f , α) =
n
i=1
f (t
i
)(α(x
i
) − α(x
i−1
)) are two Riemann sums, we have
L(P, f , α) ≤ S
1
(P, f , α) < L(P, f , α) +
and
U (P, f , α) − <S
2
(P, f , α) ≤ U (P, f , α).
Using the hypothesis that lim
d(P)→0
S(P, f , α) = A exists, we see that given
>0 there exists