Page 221 - DMTH401_REAL ANALYSIS

Unit 17: Uniform Convergence and Differentiability

Theorem 3: Differentiability for Series Notes
¥
If (a) all g are differentiable, (b) all g are continuous, (c) å n= 1 g converges uniformly and (d)
¢
¢
n
n
n
c
c $ Î [ , ] s.t. å ¥ g n ( ) < ¥ then å ¥ g converges uniformly and
a
b
n
n= 1 n= 1
æ ¥ ö ¥ ¢
çå gn ¢ = å g n .
÷
è n= 1 ø n= 1
Proof: Since sum of differentiable functions is differentiable we observe that the partial sum
¢
¢
¢
f = g + g + ... g is differentiable. Similarly, all f = g + g + ... g are continuous. By (c)
¢
+
+
2
n
1
n
2
1
n
n
¥ ¥
c
a
b
¢
¢
n å
c
,
f ® n= 1 g uniformly and by (d) c$ Î [ , ] s.t. å n= 1 g n ( ) ¥ i.e. lim f n ( )< ¥ . Therefore we
n
n®¥
¥ ¥ ¥ ¢
) å
can apply Theorem to f and observe that f ® å n= 1 g uniformly and (å n= 1 g ¢ = n= 1 g n
n
n
n
n
17.3 Central Principle of Uniform Convergence
.
Definition 2: Let { }f ¥ be a sequence of real-valued functions on I Ì  Then { }f ¥ is called
n n= 1 n n= 1
a uniform Cauchy sequence if:
"e > 0 N Î " , n m ³ N : f - f < . e
$
n m sup
Theorem 4: Central Principle of Uniform Convergence, CPUC
¥
¥
.
Let { }f n n= 1 be a sequence of real-valued functions on I Ì  Then { }f n n= 1 converges uniformly
¥
of I if and only if { }f n n= 1 is a uniform Cauchy sequence on I.
Proof: ‘’ : Suppose f converges uniformly to some f. Let e > , since f ® f uniformly we
0
n
n
have
x
x
x
$ N Î " n ³ " Î : I f n ( ) f ( ) < e /4.
-
x
Then " n , m ³ " Î : I
x
x
x
x
x
f n ( ) - f m ( ) = f n ( ) - f ( ) + f m ( ) + f ( ) £
x
x
x
x
x
£ f n ( )- f ( ) + f m ( )- f ( ) < e /4 + e /4 = e /2
Therefore,
x
f - f = sup f ( )- f ( ) £ e /2 < e .
x
n m sup n m
x Î I
’’ : Let { }f n be a uniform Cauchy sequence i.e.
"e > 0 N Î " , n m ³ N : f - f ( ) < e /2.
x
$
n m sup
In particular f n ( ) - f m ( ) < e /2 for any x Î Look at the sequence of numbers { ( )}f x ¥ n= 1
x
x
. I
n
which is usual sequence of number and hence converges. Denote its limit ( ).f x Now let m ® ¥
and get
LOVELY PROFESSIONAL UNIVERSITY 215