Page 186 - DMTH401_REAL ANALYSIS

Real Analysis

Notes The sequences of functions are denoted by (f ), (g) etc., it (f ) is a sequence of functions defined on
n n
A, then its members f , f , f ,.... are real functions with domain as the set A. These are also called
1 2 3
the terms of the sequence (f ).
n
Example: Let f (x) = x , n = 1, 2, 3,...., where x A = {x : 0  x  I}. Then (f ) is a sequence of
n
n n
functions defined on the closed interval [0, 1].
Similarly consider (f ), where f (x) = sin x, n = 1, 2, 3,.... x R. Then {f }, is a sequence of functions
n n n n
defined on the set R of real numbers.
Suppose (f ) is a sequence of functions defined on a set A and we fix a point x of A, then the
n
sequence (f (x)), formed by the values of the members of (f ), is a sequence of real numbers. This
n n
n
sequence of real numbers may be convergent or divergent. For example suppose that f (x) = x ,
n
1 1
n
x [–1, 1]. If we consider the point x = , then the sequence (f (x)) is (( ) ) which converges to 0.
2 n 2
If we take the point x = –1, the sequence (f (x)) is the constant sequence (1, 1, 1.....) which converges
n
to 1. If x = –1, the sequence (f, (x)) is (–1, I, –1, 1,....) which is divergent.
Thus, you have seen that the sequence (f (x)) may or may not be convergent. If for a sequence (f )
n n
of functions defined on a set A, the sequence of numbers (f (x)) converges for each x in A, we get
n
a function f with domain A whose value f(x) at any point x of A is lim fn(x). In this case (f ) is said
n®¥ n
to be pointwise convergent to f. We define it in the following way:
Definition 2: Pointwise Convergence
A sequence of functions (f ) defined on a set A is said to be convergent pointwise to f if for each
n
x in A, we have lim f (x) = f(x). Generally, we write f ® f (pointwise) on A.
n n
or lim f (x) = f(x) pointwise on A. Also f is called pointwise limit or limit function of (f ) on A.
n®¥ n n
Equivalently, we say that a sequence {f } converges to f pointwise on the set A if, for each > 0
n
and each xA, there exists a positive integer in depending both on  and x such that
|f (x) – f(x)| < , whenever n  m,
n
Now we consider some examples.

Example: Show that the sequence (f ) where f (x) = x , x  [0, 1] is pointwise convergent.
n
n n
Also find the limit.
n
Solution: If 0 x < 1, then lim f (x) = lim x = 0.
n®¥ n n®¥
If x = 1, then lim f(x) = lim 1 = 1.
n®¥ n®¥
Thus (f,) is pointwise convergent to the limit function f where f(x) = 0 for 0  x  1 and f(x) = 1 for
x = 1.

n
Task Show that the sequence of functions (f ) where f (x) = x , for x[–1, 1] is not pointwise
n n
convergent.

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