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Greatest lower bound (inf)
When (i) l is a lower bound for a set A of real numbers, and (ii)
every m , greater than l , is not a lower bound for A ,
then and only then, l is called the greatest lower bound of A , and
denoted by inf A .
The meanings of the terms 'greatest lower bound' and 'inf' lie in the
words of this definition, and nowhere else!
If a set of numbers has a least member, that number is the greatest
lower bound for the set.
In qn 68 you proved that 0 is the greatest lower bound of
x 0 x 1 .
In qn 70 you should havefound that 0 is the greatest lower bound of
x 0 x 1 , x 0 x 1 , 1 ( 1) / n n N and
q
q
2, q
Q
.
71 Show that a set A cannot have two different greatest lower bounds.
72 If a set A has a greatest lower bound inf A , and is any positive
number, why must there be an a A such that
inf A a (inf A ) ?
73 If l is a lower bound for the set A , and for any positivenumbr
,
you can find an a A , such that l a l , must l inf A ?( Hint .
Draw a diagram. Suppose l m and m is a lower bound for A . Try
putting m l .)
74 Let A be a set of real numbers with a least upper bound sup A , and
let B x x A . Provethat inf B sup A .
75
(i) Prove that the set of numbers x x 0 has no greatest
member.
(ii) Prove that the set of numbers x x 10 has no greatest
member.
76 List some upper bounds for the set x x 0 , and then identify the
set of all upper bounds for x x 0 . Does the set of all upper
bounds have a least member?
77 List some upper bounds for the set x x 0 , and then identify the
set of all upper bounds for this set. Does the set of upper bounds
have a least member?
78 Let ( a
) be a monotonic decreasing sequence with limit a . Prove
that a
inf
a
n N
. Useqn 66 and qn 74.
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