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Math 310 
hw
2 solutions
4.1 h, j, n, p; 4.8, 4.14 a
Friday, 11 Sept 2009
4.1 For each set below that is bounded above, list three upper bounds for the set. Otherwise write “Not bounded
above” or “NBA”.
(h)
∪
∞
n
=1
[2
n,
2
n
+ 1]
This set is not bounded above (by the Archimedean Property).
±
(j)
{
1

1
3
n
:
n
∈
N
}
Three upper bounds are 1
,
2
,
3 (note that for
n
∈
N
, we have
1
3
n
>
0 and so 1

1
3
n
<
1).
±
(n)
{
r
∈
Q
:
r
2
<
2
}
Three upper bounds are
√
2
,
√
3
,
2.
±
(p)
{
1
,
π
3
,π
2
,
10
}
Three upper bounds are 10
,
11
,
12 (note that 10 is the maximum of the set).
±
4.8 Let
S
and
T
be nonempty subsets of
R
with the following property:
s
≤
t
for all
s
∈
S
and
t
∈
T
.
(a) Observe that
S
is bounded above and that
T
is bounded below.
Since both
S
and
T
are nonempty, there are elements
s
0
∈
S
and
t
0
∈
T
. By the given property we have
s
≤
t
0
for all
s
∈
S
and
s
0
≤
t
for all
t
∈
T
. Hence,
t
0
is an upper bound for
S
and
s
0
is a lower bound for
T
. It follows that
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This note was uploaded on 03/26/2010 for the course MATHEMARIC 131a taught by Professor Janeday during the Spring '10 term at San Jose State University .
 Spring '10
 janeday

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