MAT125A Final and Solutions
March 17, 2010
1. Show that if
f
: [0
,
∞
)
→
R
is continuous and
lim
x
→∞
f
(
x
)
is finite, then
f
is bounded on [0
,
∞
).
Solution:
Let
L
= lim
x
→∞
f
(
x
). Then there exists
α >
0 such that
x > α
implies
f
(
x
)
< L
+ 1. Since
f
is continuous and [0
, α
] is compact,
f
is bounded on [0
, α
]
— say by
M
. Let
γ
= max
{
L
+ 1
, M
}
. Then

f
(
x
)
 ≤
γ
for all
x
≥
0.
2. Prove that the metric space (
X, d
) where
X
=
R
and
d
:
X
×
X
→
[0
,
∞
) defined
by
d
(
x, y
) =
(
0
x
=
y
1
x
6
=
y
is complete. That is, show that every Cauchy sequence
{
x
n
}
in (
X, d
) converges to
a point
x
∈
X
.
Solution:
Let
{
x
n
}
be a Cauchy sequence in
X
. That means that for all
>
0
there exists
N
such that
d
(
x
n
, x
m
)
<
for
n, m > N
. In particular, there exists an
N
such that
d
(
x
n
, x
m
)
<
1
/
2 for all
n, m > N
. But, by the definition of the metric,
d
(
x
n
, x
m
)
<
1 implies
x
n
=
x
m
. So the sequence
{
x
n
}
is constant for
n > N
and
hence convergent.
3. Show that if
{
a
k
}
is a sequence of positive real numbers such that
(1)
∞
X
k
=1
a
k
<
∞
,
then
(2)
∞
X
k
=1
ln(1 +
a
k
)
<
∞
.
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 Winter '10
 Bremer
 Calculus, Mean Value Theorem, Continuous function, Xn, uniformly convergent limits

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