Math 318 HW #4 Solutions
1. Abbott Exercise 7.4.6. Review the discussion immediately preceding Theorem 7.4.4.
(a) Produce an example of a sequence
f
n
→
0 pointwise on [0
,
1] where lim
n
→∞
1
0
f
n
does
not exist.
Answer.
Consider the function
f
n
whose graph is given in Figure
1
.
n
2
1
/n
2
/n
Figure 1: The function
f
n
For any
x
∈
[0
,
1], either
x
= 0 and
f
n
(
x
) = 0 for all
n
, or
f
n
(
x
) = 0 for all
n >
2
/x
;
either way,
f
n
(
x
)
→
0, so we see that
f
n
→
0 pointwise.
On the other hand,
1
0
f
n
=
1
2
2
n
·
n
2
=
n,
so lim
n
→∞
1
0
f
n
diverges to +
∞
.
(b) Produce another example (if necessary) where
f
n
→
0 and the sequence
1
0
f
n
is un
bounded.
Answer.
The same example from (a) still works.
(c) Is it possible to construct each
f
n
to be continuous in the examples of parts (a) and (b)?
Answer.
The example in (a) is already continuous.
(d) Does it seem possible to construct the sequence (
f
n
) to be uniformly bounded?
Answer.
Certainly we can’t get
1
0
f
n
to be unbounded if the
f
n
are uniformly bounded:
if there’s a uniform bound
M
such that

f
n
(
x
)
 ≤
M
for all
n
and all
x
∈
[0
,
1], then
1
0
f
n
≤
M
(1

0) =
M
for any
n
.
Also, problem 2 below implies it is not possible to have
1
0
f
n
diverge at all when the
convergence is uniform on any set of the form [
δ,
1] (as it would be for, say, a power
series centered at
x
= 1).
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '08
 Staff
 Math, Topology, Trigraph, Open set, dα, gn

Click to edit the document details