Page 93 - DMTH401_REAL ANALYSIS

Unit 7: Convergent Sequence

Notes
(b) If K is a sequence of compact sets in X such that K  K (n  ) and if lim diam K = 0
n n n+1 n¥ n
¥
then  1 K consists of exactly one point.
n
7.7 Cauchy Sequences and Convergent Sequences

Theorem:

(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 to
n n
some point of X.
k
(c) In  every Cauchy sequence converges.
7.7.1 Complete Spaces

Definition:

A metric space is said to be complete if every Cauchy sequence converges.
k
Notice that all compact metric spaces are complete but there are metric spaces (like  ) which are
complete but not compact.
Lemma
Every closed subset of a complete metric space is complete.

7.8 Increasing/Decreasing Sequences

Definition:
A sequence {Sn} of real numbers is said to be
(a) Monotonically increasing if S  S for all n 
n n+1
(b) Monotonically decreasing if S  S for all n 
n n+1
(c) Monotonic if it is monotonically increasing or monotonically decreasing.

Theorem: Suppose {S } is monotonic. Then {S } converges if and only if {S } is bounded.
n n n
Self Assessment

Fill in the blanks:
1. If there is any ambiguity we say {p } ........................ in X.
n
2. 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 ........................ .
3. If {p } is a sequence in a ........................ space X, then some subsequence of {p } converges to
n n
a point of X.
4. Every bounded sequence in  contains a ........................ .
k
5. A metric space is said to be complete if every ........................ .

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