Page 270 - DMTH401_REAL ANALYSIS

Real Analysis

Notes Consequently, if P = {x , x , … x } be any partition of [a,b]
0 1 n
and m, be the inf. of h in [x, 1, x,], then
i
m  " = 1, 2,¼ n
0
i
n
Þ å m D x  0
i
i
=
i 1
Þ L(P,h) 0

Thus for every partition P, the lower sum L(P,h)  0.
In other words, Sup. (1 (P,h): P is a partition of [a,b])  0
or
h
ò f(x) dx
a
Since h is integrable in [a,b], therefore

b b b
ò h(x) dx = ò h(x) dx = ò h(x) dx.
a a a
Thus
b
ò h(x) dx  0
a
or

b
ò (g - f) (x) dx  0
a

b b
Þ ò g(x) dx  ò f(x) dx
a a
which proves the property
Property 2: If f, is integrable on [a, b] then f is also integrable on

b b
[a,b] and f(x) dx £ ò f(x) dx
ò
a a
Proof: The inequality follows at once from Property 1 provided it is known that f is integrable
on [a,b]. Indeed, you know that - f £ £ f .
f
Therefore,

b b b
ò f(x) dx £ ò f(x) dx £ ò f(x) dx
- a a

which proves the required result. Thus, it remains to show that f is integrable.
Let Î > 0 be any number. There exists a partition P of [a, b]

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