Graphics Reference
In-Depth Information
The method used in qn 84 applies quite generally and provides a more
illuminating way of describing lim sup
a
and lim inf
a
. This description
enables us to prove, for example, that a sequence (
a
) is convergent if
and only if lim sup
a
.
In the absence of completeness, a bounded sequence need not have
any convergent subsequences (take the infinite decimal sequence of
2
for example) and therefore neither lim sup nor lim inf may be defined.
lim inf
a
Summary
-
completeness
The completeness principle
Every infinite decimal sequence is convergent.
Definition
The limit of every infinite decimal sequence is a
real number
heorem
qns 34, 35
T
Every bounded monotonic sequence is
convergent.
Theorem
qn 36
The sequence with
n
th term (1
1/
n
)
is
convergent with limit (e) between 2 and 3.
Theorem
qn 40
Every positive real number has a unique
n
th
root.
Every bounded sequence has a convergent
subsequence.
Definition
A sequence (
a
Theorem
qn 46
) is said to satisfy the
Cauchy
criterion
, when, given
0, there exists an
N
,
such that
n
N
a
a
, for all
positive integers
k
.
Definition
A sequence satisfying the Cauchy criterion is
called a
Cauchy sequence
.
The General Principle of Convergence
qns 55
—
57 A sequence is convergent if and only if it is a
Cauchy sequence.
Definition
An upper bound for a set
A
of real numbers is
called a least upper bound, when no lesser
number is an upper bound for the set. The least
upper bound of
A
is denoted by sup
A
.
Theorem
qn 80
Every non-empty set of real numbers which is
bounded above has a least upper bound.
Theorem
Any oneof thefollowing propositions is
suMcient to imply the completeness principle:
After qn 35
1. Every bounded monotonic sequence is
convergent.
After qn 46
2. Every bounded sequence has a convergent
subsequence.
