Graphics Reference
In-Depth Information
Least upper bound (sup)
sup
sup
sup
When (i)
u
is an upper bound for a set
A
of real numbers, and (ii)
every
v
less than
u
is not an upper bound for
A
, then and only then,
u
is called the
least upper bound
of
A
, and denoted by sup
A
.
Themeanings of the terms 'least upper bound' and 'sup' lie in the
words of this definition, and nowhere else!
If a set of numbers has a greatest member, the greatest member is
the
least upper bound
of that set.
In qn 60 you proved that 1 is the
least upper bound
of
x
0
x
1
, and that 3 is the
least upper bound
of
x
0
x
1, 2
x
3
. A finite set of real numbers and a closed
interval are examples of sets with a greatest member.
In qn 61 you should havefound that 2 is the
least upper bound
of
. A strictly increasing sequence
and an open interval are examples of sets without a greatest member.
The
least upper bound
is then a cluster point of the set. This will follow
from qn 64.
2
1/
n
n
N
and
x
x
2,
x
Q
62 If a set
A
has a least upper bound, sup
A
, must sup
A
be a member
of
A
? Might sup
A
be a member of
A
?
63 Show that a set
A
cannot have two different least upper bounds.
64 If a set
A
has a least upper bound sup
A
, why must there be an
a
A
such that (sup
A
)
a
sup
A
, for any positivenumbr
?
Use this result to construct a strictly increasing sequence of
elements of
A
within theintrval (sup
A
1, sup
A
) on the
assumption that sup
A
A
.
65 If
u
is an upper bound for the set
A
, and for any positivenumbr
,
you can find an
a
A
, such that
u
a
u
, must
u
sup
A
?
