Page 92 - DMTH401_REAL ANALYSIS

Real Analysis

Notes (b) Suppose {x }, (y ) are sequence in  , { } is a sequence of real numbers, and x  x, y  y,
k
n n n n n
  . Then
n
lim (x + y ) = x + y
n¥ n n
lim (x  y ) = x  y
n¥ n n
lim  x = x
n¥ n n
7.2.1 Subsequences

Definition:
Given a sequence {p }, consider a sequence {n } of positive integers such that n < n < n . . . Then
n k 1 2 3
the sequence {p } is called a subsequence of {p }. If {p } converges its limit is called a subsequential
ni n ni
limit of {p }.
n
It is clear that {p } converges to p if and only if every subsequence of {p } converges to p.
n n
7.3 Subsequences and Compact Metric Spaces

Theorem:
(a) If {p } is a sequence in a compact metric space X, then some subsequence of {p } converges
n n
to a point of X.
k
(b) Every bounded sequence in  contains a convergent subsequence.
7.4 Subsequences Limits

Theorem:
The subsequential limits of a sequence {p } in a metric space X form a closed subset of X.
n
7.5 Cauchy Sequence

A sequence {p } in a metric space (X, d) is said to be a Cauchy sequence if for every  > 0 there is
n
an integer N such that d(p , p ) <  for all n, m  N.
n m
Definition:
Let E be a non-empty subset of a metric space (X, d), and let S = {d(p, q) : p, q  E}. The diameter
of E is sup S.
If {p } is a sequence in X and if E consists of the points p , p , . . ., it is clear that {p } is a Cauchy
n n N N+1 n
sequence if and only if
lim diam E = 0
N¥ N
7.6 Cauchy Sequence and Closed Sets

Theorem:

(a) If E is the closure of a set E in a metric space X, then

diam E = diam E

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