Math 142A
(P. Fitzsimmons)
Second Midterm Solutions
November 14, 2001
1.
(a) A real number
x
is a
cluster point
of a sequence
{
a
n
}
if for each
>
0 there are infinitely many indices
n
such that

x
−
a
n

<
.
(b) The
n
th
partial sum
of
∑
∞
k
=1
a
k
is the finite sum
s
n
=
∑
n
k
=1
a
k
.
(c) The composition
f
◦
g
of two functions
f
and
g
with respective domains
D
f
and
D
g
is the function
defined by the formula (
f
◦
g
)(
x
) =
f
(
g
(
x
)) on the domain
D
f
◦
g
consisting of those real numbers
x
∈
D
g
for
which
g
(
x
)
∈
D
f
(so that
f
(
g
(
x
)) is well defined).
2.
Let
{
a
n
}
∞
n
=1
be a Cauchy sequence of real numbers. Let
f
:
R
→
R
be a function such that

f
(
x
)
−
f
(
y
)
 ≤

x
−
y

for all real
x
and
y
. Prove that the sequence
{
f
(
a
n
)
}
∞
n
=1
is also a Cauchy sequence.
Solution.
Let
>
0 be given. Because
{
a
n
}
is a Cauchy sequence, there is a cutoff
N
=
N
( ) such that

a
m
−
a
n

<
if
m,n > N
. But then by the assumed property of
f
, we have

f
(
a
m
)
−
f
(
a
n
)
 ≤ 
a
m
−
a
n

<
if
m,n > N.
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 Fall '01
 Fitzsimmons
 Math, Calculus, Mathematical Series, Cauchy sequence

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