Page 260 - DMTH401_REAL ANALYSIS

Real Analysis

Notes Î Î
= [U(T,f) – L(T,f)] + [U(T,g) – L(T,g)] < + = Îfor T occurring in (5). This shows that f + g Î R(a,b)
2 2
b b b
It remains to show that [f(x) g(x)]dx+ =  [f(x) dx +  g(x)

a a a
Now
b b
+
£
+
+
+
£
 (f g)(x)dx =  (f g)(x) dx U(P,f g) U(P,f) U(P,g) (6)
a
for any partition P of [a,b]. Given any Î > 0 we can find a partition P of [a,b] such that
b
U(P,f) <  f(x)(x) dx+ Î /2
a
b
U(P,g) <  g(x) dx+ Î /2 (7)
a
Substituting (7) in (6), we obtain
b b b
+
 (f g)(x) dx <  f(x) dx +  g(x) dx+ Î (8)
a a a
Since (8) holds for arbitrary Î > 0, we obtain
b b b
+
 (f g)(x) dx £  f(x) dx +  g(x) dx (9)
a a a
Replacing f and g by –f and –g in (9) we obtain
b b b
-
-
-
-
 ( f g)(x) dx £  { f (x)} dx +  { g(x)} dx
or
b b b
+
 (f g) (x) dx £ -  f(x) dx -  g(x) dx
a a a
This is equivalent to
b b b
+
 (f g) (x) dx   f(x) dx +  g(x) dx
a a a
Combining (9) and (10), we get
b b b
+

 (f g) (x) dx =  f(x) dx g(x) dx
a a a
Which proves the theorem.
Theorem 12: If f Î R(a,b) and g Î R (a,b), then f – g Î R(a, b) and

b b b
-
 (f g) (x) dx =  f(x) dx -  g(x) dx.
a a a
Proof: Since g E R [a,b], therefore -g Î R [a,b] and
b b
 - [g(x)]dx = -  g(x) dx

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