Page 94 - DMTH401_REAL ANALYSIS

Real Analysis

Notes 7.9 Summary

 A sequence {p } in a metric space (X, d) is said to converge if there is a point p  X with the
n
following property:
( " > 0)(N) ( " n > N) d(p , p) < 
n
 In this case we also say that {p } converges to p or that p is the limit of {p } and we write
n n
p  p or lim limp = p .
n n
n¥
 If {p } does not converge we say it diverges.
n
 If there is any ambiguity we say {p } converges/diverges in X.
n
 The set of all p is said to be the range of {p } (which may be infinite or finite). We say {p }
n n n
is bounded if the range is bounded.
 Properties of convergent sequences
(a) {p } converges to p  X if and only if every neighbourhood of p contains p for all but
n n
finitely many n.
(b) If p, p’  X and if {p } converges to p and to p’ then p = p’
n
(c) If {p } converges then {p } is bounded.
n n
(d) If E  X and if p is a limit point of E, then there is a sequence {p } in E such that
n
p = limp n
n¥
 Theorem of couchy sequences and convergent sequences
(a) In any metric space X, every convergent sequence is a Cauchy sequence.
(b) If X is a compact metric space and if {p } is a Cauchy sequence in X then {p } converges
n n
to some point of X.
k
(c) In  every Cauchy sequence converges.
7.10 Keywords

Subsequential: Given a sequence {p }, consider a sequence {n } of positive integers such that
n k
n < n < n . . . Then the sequence {p } is called a subsequence of {p }. If {p } converges its limit
1 2 3 ni n ni
is called a subsequential limit of {p }.
n
Subsequential Limits: The subsequential limits of a sequence {p } in a metric space X form a
n
closed subset of X.
Cauchy Sequency: A sequences {p } in a metric space (X, d) is said to be a Cauchy sequency if for
n
every  > 0 there is an integer N such that d(p , p ) <  for all n, m  N.
n m
7.11 Review Questions

1. Define convergent sequence.
2. Discuss the properties of convergent sequence.
3. Explain subsequences and compact metric spaces.
4. Describe subsequence limits.

5. Explain the Cauchy sequences and convergent sequences.

88 LOVELY PROFESSIONAL UNIVERSITY