Page 206 - DMTH401_REAL ANALYSIS

Real Analysis

Notes
Figure 15.6: Claim 1 (left) and
Claim 2 (right) visualizations.

15.4 Dini’s Theorem

Now, we are above to state in certain sense a converse to Theorem 1.

Theorem 4: (Dini’s Theorem). Let { }f ¥ be a sequence of real-valued functions on [ , ].a b If:
n n= 1
(a) f  f pointwise, (b) all f are continuous, (c) f is continuous and (d) x" Î [ , ] : { ( )} ¥ n= 1
a
x
b
f
n
n
n
is monotone then f  f uniformly.
n
0
Proof: Given e > we want
a
x
x
"e > 0 N Î " n ³ " Î [ , ] : f n ( ) - f ( ) < e .
$
x
b
b
a
Let x Î [ , ]. Since f  f pointwise
n
x
x
x
$ N ( ) Î " n ³ N ( ) : f n ( ) < e /2.
In particular,
x
x
f N ( ) ( ) - f ( ) < e /2.
x
y
y
Let ( )g y = f N ( ) ( )- f ( ). Then ( )g y is continuous by (b) and (c). In particular ( )g y is continuous
x
at x
x
a
-
b
"
-
$d ( ) > 0 . . y Î [ , ] : y x < d ( ) Þ g ( ) g ( ) < e /2
s
y
x
x
t
which implies
y
x
y
y
f N ( ) ( ) - f ( ) = g ( ) £ g ( ) g ( ) + g ( ) =
-
y
x
x
y
-
x
= g ( ) g ( ) + f ( )- f ( ) < e /2+ e /2 < e .
x
x
N ( )
x
Moreover,
x
-
y
a
"
" n ³ N ( ) y Î [ , ] : y x < d ( ) Þ f ( )- f ( ) < e
b
y
x
n
since, by (d)
y
y
f n ( ) - f ( ) £ f N ( ) ( ) - f ( ) .
y
y
x
a
x
Denote ( ) (I x = x - d ( ), x+ d ( )) x Î [ , ]. We have shown
"
x
b
y
x
b
y
x
$ N ( ) Î " n ³ N ( ) y Î I ( ) Ç [ , ] : f ( ) - f ( ) < e .
"
a
x
n
Note that, { ( )}I x x Î [ , ] is a cover for [ , ],a b since x" is covered at least by ( ).I x By Heine-Borel
a
b
Theorem 3 there is finite subcover { ( ), ( ),..., (I x 1 I x 2 I x m )}. Choose N = max{ ( ), ..., N x m )} for
(
x
N
1
I
x
b
x
a
any e > 0. Let n ³ N and x Î [ , ]. Let ( ) be an interval converging x. Then x x- i < d ( ).
i
i
x
x
x
Since n ³ N we have n ³ N ( ) and therefore f n ( ) - f ( ) < e .
i
Are all conditions of Dini’s Theorem 4 important? The answer is yes, look at following examples.
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