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MAT 371
Advanced Calculus
/120
September 30, 2004
Test 1
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1. a.
State the
least upper bound axiom
(or
supremum axiom
).
b.
State “the” (“a” ?) deﬁnition of
limit point
.
c.
State “the” (“a” ?) deﬁnition of
open set
.
2. a.
Using only the ﬁeld axioms, prove
ONE
of the following:
(i)
∀
x
∈
R
,
0
·
x
= 0
,
or
(ii)
∀
x
∈
R
,
(

1)
·
x
=

x
.
If impossible explain why.
b.
Using only the ﬁeld axioms, prove that

1
6
= 1
If impossible explain why.
3.
Suppose that
{
I
i
= [
a
i
, b
i
]:
i
∈
Z
+
}
is a collection of nested
nontrivial
intervals in
R
, i.e.,
for all
i
∈
Z
+
,
a
i
≤
a
i
+1
< b
i
+1
≤
b
i
. Prove that the intersection
T
∞
i
=1
I
i
is nonempty.
Bonus
Consider the sequences (
a
n
)
∞
n
=1
and (
b
n
)
∞
n
=1
of
rational
(!) numbers deﬁned
by
a
n
= (1 +
1
n
)
n
and
b
n
= (1 +
1
n
)
n
+1
.
Show that (
a
n
)
∞
n
=1
and (
b
n
)
∞
n
=1
are monotonically increasing and decreasing, respectively.
(Hint: Analyze the quotients
a
n
+1
a
n
and
b
n
+1
b
n
.) Also show that both sequences are bounded.
What can you conclude about the convergence of the sequences?
4. a.
Suppose that
S
⊆
R
is a subset of the set of real numbers with supremum
z
= sup
S
.
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This note was uploaded on 04/02/2009 for the course MAT 371 taught by Professor Thieme during the Spring '07 term at ASU.
 Spring '07
 thieme
 Calculus

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