20
CHAPTER 2. METRIC SPACES AND TOPOLOGY
P2
.1
.4
.
Let
x
=(
x
1
,...,x
n
)
,y
=(
y
1
,...,y
n
)
∈
R
n
and consider the function
ρ
given by
ρ
(
x
,y
)
=max
{
x
1

y
1

,...,

x
n

y
n
}
.
Show that
ρ
is a metric.
P2.1
.5
.
Let
X
be a metric space with metric
d
.DeFne
¯
d
:
X
×
X
→
R
by
¯
d
(
x,y
)=m
in
{
d
(
x,y
)
,
1
}
.
Show that
¯
d
is also a metric.
Let
(
X,d
)
be a metric space. Then, elements of
X
are called
points
and the
number
d
(
x,y
)
is called the
distance
between
x
and
y
.Le
t
$>
0
and consider the
set
B
d
(
x,$
)=
{
y
∈
X

d
(
x,y
)
<$
}
.Th
i
sse
ti
sca
l
ledthe
d
open ball
of radius
$
centered at
x
.
P2
.1
.6
.
Suppose
a
∈
B
d
(
x,$
)
with
$>
0
.S
h
owt
h
a
tt
h
e
r
ee
x
i
s
t
sa
d
open ball
centered at
a
of radius
δ
,say
B
d
(
a,δ
)
,thatiscontainedin
B
d
(
x,$
)
.
One of the main beneFts of having a metric is that it provides some notion of
“closeness” between points in a set. This alows one to discusss limits, convergence,
open sets, and closed sets.
Defnition 2.1.7.
A
sequence
of elements from a set
X
is an inFnite list
x
1
,x
2
,...
where
x
i
∈
X
for all
i
∈
N
.±
o
rm
a
l
l
y
,as
e
q
u
e
n
c
e
i
se
q
u
i
v
a
l
e
n
t
t
oa
f
u
n
c
t
i
o
n
f
:
N
→
X
where
x
i
=
f
(
i
)
for all
i
∈
N
.
Defnition 2.1.8.
Consider a sequence
x
1