18.100C Presentation topics for 9/10
Problem 1
(Rudin Problems 4,5 pg.22)
Review the notion of upper and lower
bound, inﬁmum, and supremum, then solve:
(i) Let
E
⊂
S
be a nonempty subset of an ordered set
S
. Suppose
α
∈
S
is a
lower bound of
E
and
β
∈
S
is an uppper bound of
E
. Prove that
α
≤
β
.
(ii) Let
A
⊂
R
be a nonempty subset of the real numbers.
Deﬁne

A
=
{
x
ﬂ
ﬂ
x
∈
A
}
to be the set of all numbers

x
, where
x
∈
A
.
Show that inf
A
=

sup(

A
).
(Consider separately the cases when
A
is bounded below and when not.)
Problem 2
(Rudin Problem 9 pg.22 – lexicographic order)
Review the notion
of ordered sets and leastupperbound property, then solve:
For complex numbers
z
=
a
+
bi
∈
C
and
w
=
c
+
di
∈
C
deﬁne “
z < w
” if
either
a < c
or if (
a
=
c
and
b < d
). Prove that this turns
C
into an ordered
set. Is this an ordered ﬁeld? Does it have the leastupperbound property?
Problem 3
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 Fall '10
 Prof.KatrinWehrheim
 nonempty subset, leastupperbound property

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