MATH 74 HOMEWORK 10  DUE MONDAY, APRIL 30
In problems 14 we continue the notation from last week: we have a function
L
: (0
,
∞
)
→
R
satisfying (1)
L
(10) = 1, (2) if 0
< x < y
then
L
(
x
)
< L
(
y
), and
(3)
L
(
xy
) =
L
(
x
) +
L
(
y
) for all
x, y
∈
(0
,
∞
). Feel free to use things proved on last
week’s homework (the problems and solutions are online).
For
n
∈
N
, define
a
n
to be the unique number in
N
0
satsifying
10
a
n
≤
30
n
<
10
a
n
+1
.
For example since 10
1
≤
30
1
<
10
2
, we have
a
1
= 1, and since 10
2
≤
30
2
<
10
3
, we
have
a
2
= 2.
1.
What are
a
3
and
a
4
?
There is a relationship between
a
n
and the number of
decimal digits of 30
n
. What is it?
2.
Fix
n
∈
N
. Prove that
a
n
n
≤
L
(30) and that
L
(30)
<
a
n
+1
n
.
3.
Define for
n
∈
N
the sequences
b
n
=
a
n
n
and
c
n
=
a
n
+1
n
.
(a) Is the sequence
b
n
bounded above?
(b) Is the sequence
b
n
increasing? [Note that
a
n
is.]
(c) Is the sequence
c
n
bounded below?
(d) Is the sequence
c
n
decreasing?
4.
Prove that the sequences
b
n
and
c
n
are both convergent to
L
(30). [You may use
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 Fall '07
 COURTNEY
 Math, triangle, Limit of a sequence, Sequence space, bn bn cn

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