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81 Articulate a theorem for greatest lower bounds analogous to that of
qn 80.
82 If A and B denote bounded sets of real numbers, how do the
numbers sup A , inf A , sup B and inf B relate if B A ? Give
examples of unequal sets for which sup A
sup B and inf A
inf B .
limsup and liminf
The ideas in questions 83 and 84 will only be used in later chapters
in qn 5.102 and 5.107. 5.107 will be used at the end of chapter 12.
They have been bracketed, not because they are incidental in the
development of analysis, but because they may most profitably be
studied at a second reading.
(83)For any bounded sequence with an upper bound U and lower
bound L
(i) how do you know that ( a
) contains a convergent
subsequence;
(ii) if a is the limit of a convergent subsequence, how do you
know that L a U ;
(iii) how do you know the set of limits of convergent subsequences
has a supremum and an infimum?
The supremum is called limsup a n and theinfimum is called
liminf a n
Question 83 shows the power of the theorems we have developed by
establishing the existence of numbers about which we know very little.
A concrete and constructive approach to the same concepts is given in
thefollowing qustion.
(84)
(i) Illustrate on a graph the first few terms of the sequence with
n th term a
( 1) (1 1/ n ).
(ii) Is this sequence bounded?
(iii) Is this sequence convergent?
(iv) Identify one convergent subsequence.
(v) Find sup a
n N and inf a
n N .
(vi) Let u
sup a
k n and let l
inf a
k n . Find the
first four terms of the sequence ( u
) and thefirst four trms of
the sequence ( l
).
(vii) Explain why u
u
for all k , and why l
l
for all k .
(viii) Areboth ( u
) bounded monotonic sequences?
(ix) The limits of these sequences are lim sup a
) and ( l
and lim inf a
respectively. Find these limits.
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