4130 HOMEWORK 6
Due Thursday April 1
(1) Let
A
⊂
R
. A point
x
∈
A
is called
isolated
if it is not a cluster point of
A
.
(a) Can an open set have an isolated point? Can a closed set have one?
(b) Give an example of a countable set with no isolated points.
(2) Section 3.3.1 # 8.
(3) Section 4.2.4 # 3. (Recall that an
interval
is, by deﬁnition, a subset
I
of
R
such that
for all
x,y
∈
I
and all
z
∈
R
with
x < z < y
, we have
z
∈
I
.)
(4) In this question, we will show that every positive real number has an
n
th
root.
(a) Let
x
∈
(0
,
∞
) and
n
∈
N
. Show that there exist
α,β
∈
R
with
α
n
< x < β
n
.
(b) Show that there exists
y
∈
R
with
x
=
y
n
.
(c) For
x
∈
[0
,
∞
), show that there exists a
unique
y
∈
[0
,
∞
) with
x
=
y
2
. We
denote this
y
by
√
x
.
(d) Deﬁne
f
: [0
,
∞
)
→
R
by
f
(
x
) =
√
x
. Show that
f
is a continuous function.
(5) Two monasteries,
A
and
B
, are joined by exactly one path
AB
which is 20 miles
long. One morning, Brother Albert (a monk) sets out from monastery
A
at 9 am,
arriving at monastery
B
at 9 pm. The next day, he sets out from monastery
B
at
9 am, arriving at monastery
A
at 9 pm. On both journeys, he may have stopped to
rest, or even walked backwards for some of the time.
(a) Prove that there is a point
x
on the path
AB
such that Brother Albert was at
x
at exactly the same time on both days.
(Hint: let
f
i
(
t
) be the distance of Brother Albert from
A
at time
t
on day
i
,
i
= 1
,
2. Apply the Intermediate Value Theorem to a suitable combination of
f
1
and
f
2
.)
(b) Another monk, Brother Gilbert, has been dabbling in forbidden knowledge.
Once per day, by snapping his ﬁngers, he can instantaneously teleport him-
self to any point within a 3 ft. radius of his current location. Suppose Brother
Gilbert makes the same journey as Brother Albert. Does the conclusion from
part (a) still hold?
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