This document discusses sequences and series of functions. It defines pointwise and uniform convergence of sequences of functions. Pointwise convergence means the sequence converges for each value, while uniform convergence means the convergence is uniform across all values. Cauchy's criterion for uniform convergence is presented. Theorems are provided about uniform convergence implying continuity and uniform convergence of series if partial sums are bounded. The definition of uniform convergence on compact metric spaces using the supremum norm is also given.

1. SEQUENCES AND SERIES OF FUNCTIONS
Prof. Mini Thomas
Department of Mathematics

2. CONTENT
 INTRODUCTION
 POINTWISE CONVERGENCE AND UNIFORM
CONVERGENCE OF SEQUENCES OF FUNCTIONS
 CAUCHYʹ S CRITERION FOR UNIFORM CONVERGENCE
 THEOREMS
 UNIFORM CONVERGENCE AND CONTINUITY
 DEFINITION
 CONCLUSION
 REFERENCES

3. INTRODUCTION
 In Mathematics , real analysis is the branch of
mathematical analysis that studies the behavior of real
numbers ,sequences and series of real numbers and
real valued functions .Some particular properties of
real –valued sequences and functions that real analysis
studies include convergence , limits , continuity ,
smoothness , differentiability and integrability .

4. POINTWISE CONVERGENCE AND UNIFORM
CONVERGENCE OF SEQUENCES OF FUNCTIONS
 Definition ( Pointwise convergence )
Suppose {fn} be a sequence of functions defined on a set E and
suppose that the sequence of numbers {fn₍ₓ₎} converges for every
xϵE we can define a function f by
f₍ₓ₎= limn→∞ fn₍ₓ₎ , xϵE
And we say that {fn} → f pointwise on E or simply , {fn}
converges on E and that f is the limit or the limit function of
{fn}.
⦁Definition (Uniform convergence)
A sequence of functions {fn} is said to converge uniformly on E
to a function if for every ɛ˃ 0 , their exist N such that
n ≥N implies │fn₍ₓ₎−f₍ₓ₎│≤ɛ , for all xϵE

5. CAUCHY ‘S CRITERION FOR
UNIFORM CONVERGENCE
 THEOREM : The sequence of functions {fn} defined on
E , converges uniformly on E iff for every ɛ˃0 their exist
an integer N such that
m,n ≥N , xϵE implies │fn₍ₓ₎─ fm₍ₓ₎│≤ɛ

6. THEOREMS
 THEOREM 1: Suppose limn→∞ fn₍ₓ₎ = f₍ₓ₎ , xϵE put
Mn = xϵEsup│fn₍ₓ₎─f₍ₓ₎│ then fn→f uniformly on E iff
Mn→0 as n→∞
 THEOREM 2 (WEIERSTRASS M-TEST FOR UNIFORM
CONVERGENCE OF SERIES ) :
Suppose {fn} is a sequence of functions defined on E and
suppose │fn₍ₓ₎│≤Mn , for all xϵE and n= 1,2,3,…… then
∑fn converges uniformly on E if ∑Mn converges

7. UNIFORM CONVERGENCE AND
CONTINUITY
 THEOREM
Suppose fn converges to f uniformly on a set E in a metric
space . Let x be a limit point of E and suppose that
limt→x fn ₍t₎= An ,n=1,2,3,. then {An} converges and
limt→x f₍t₎ = limn→∞ An

8.  THEOREM : If {fn} is a
sequence of continuous
functions on E and if fn
converges to f uniformly on
E then f is continuous on E
 THEOREM : suppose k is
compact
a) {fn} is a sequence of
continuous functions on
k
b) {fn} converges pointwise
to a continuous function f
on k
c) fn₍ₓ₎ ≥ fn+1₍ₓ₎ , for every
xϵk , n =1,2,… then fn
converges to f uniformly
on k

9. DEFINITION
Let X be a metric space then the set of all complex valued
continues bounded functions with domain x is denoted
by Ҫ₍ₓ₎ . If f is a continuous function on a compact
metric space then f is bounded . So if x is compact Ҫ₍ₓ₎
consists of complex valued continuous functions on X .
Let fϵҪ₍ₓ₎ then define ǁ f ǁ = xϵXsup │f₍ₓ₎│, then ǁ ǁ is a
norm on Ҫ₍ₓ₎ and is called supremum norm

10. CONCLUSION
The difference between the uniform convergence and
pointwise convergence is that if {fn} converges
uniformly on E which is possible for each ԑ>0 to find
one integer N which will do for all xϵE so in case of
uniform convergence N depends only on ԑ and in case
of pointwise convergence N depends on ϵ and x

11. REFERENCES
WALTER RUDIN, Principles Mathematical analysis (Third
edition) , McGraw Hill Book Company , International
Editions.
Chapter 7 Section 7.1- 7.18