MATH 3283W. Sequences, Series, and Foundations:
Writing Intensive. Spring 2009
Homework 3. Problems and Solutions
I. Writing Intensive Part
1
. Let
f
(
x
) be a continuous function on the segment [0
,
1]. Show that
f
is
uniformly continuous
on [0
,
1]
,
which means:
∀
ε >
0
,
∃
δ >
0
,
such that
from
x, y
∈
[0
,
1] and

x

y

< δ
it follows

f
(
x
)

f
(
y
)

< ε.
Solution.
Suppose this statement is false. Then
∃
ε >
0 such that
∀
δ >
0
there are
x, y
∈
[0
,
1] with

x

y

< δ
such that

f
(
x
)

f
(
y
)
 ≥
ε.
Choose
a sequence 0
< δ
j
→
0 as
j
→ ∞
,
and the corresponding
x
j
, y
j
∈
[0
,
1] with

x
j

y
j

< δ
j
such that

f
(
x
j
)

f
(
y
j
)
 ≥
ε.
By the Bolzano–Weierstrass
theorem, the sequence
{
x
j
}
contains a convergent subsequence:
x
j
k
→
x
0
∈
[0
,
1] as
k
→ ∞
.
Then also
y
j
k
→
x
0
as
k
→ ∞
.
Since
f
(
x
) is continuous at
x
0
,
we then have
f
(
x
j
k
)

f
(
y
j
k
)
→
f
(
x
0
)

f
(
x
0
) = 0
,
so that

f
(
x
j
k
)

f
(
y
j
k
)

< ε
for large enough
k.
This contradiction shows that the given statement is true.
2
. From Problem 6 in Homework 1 it follows
a
n
=
1 +
1
n
¶
n
< e < b
n
=
1 +
1
n
¶
n
+1
for all
n
∈
N
.
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 Spring '09
 Math, Mathematical analysis, Tn, Limit of a sequence

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