18.100B,
Fall
2002,
Homework
6
Due
by
Noon,
Tuesday
October
29.
Rudin:
(1)
Chapter
4,
Problem
20
If
E
is
a
nonempty
subset
of
a
metric
space
X,
deﬁne
the
distance
from
x
∈
X
to
E
by
ρ
E
(
x
)=
inf
d
(
x, z
)
.
z
∈
E
¯
(a)
Prove
that
ρ
E
(
x
)
=
0
if
and
only
if
x
∈
E.
(b)
Prove
that
ρ
E
is
uniformly
continuous
on
X
by
showing
that

ρ
E
(
x
)
−
ρ
E
(
y
)
≤
d
(
x, y
)
for
all
x, y
∈
X.
Solution.
(a)
If
ρ
E
(
x
)
=
0
then
there
exists
a
sequence
z
n
∈
E
such
that
¯
d
(
x, z
n
)
→
0
.
This
implies
z
n
→
x
and
hence
x
∈
E.
Conversely
if
¯
x
∈
E
then
either
x
∈
E,
in
which
case
ρ
E
(
x
)=0
,
or
else
x
∈
E
±
,
so
there
exists
a
sequence
z
n
∈
E
with
z
n
→
x.
This
implies
d
(
x, z
n
)
→
0
so
ρ
E
(
x
)=0
.
(b)
If
x, y
∈
X
then
for
any
z
∈
E,
using
the
triangle
inequality
ρ
E
(
x
)
≤
d
(
x, z
)
≤
d
(
x, y
)+
d
(
y, z
)
.
Taking
the
inﬁmum
over
z
∈
E
on
the
righthand
side
shows
that
ρ
E
(
x
)
−
ρ
E
(
y
)
≤
d
(
x, y
)
.
Interchanging
the
roles
of
x
and
y
gives
the
desired
estimate

ρ
E
(
x
)
−
ρ
E
(
y
)
≤
d
(
x, y
)
.
This
proves
the
uniform
continuity
of
ρ
E
,
since
given
±>
0
,d
(
x, y
)
<±
implies

ρ
E
(
x
)
−
ρ
E
(
y
)

<±.
(2)
Chapter
4,
Problem
23
Area
lva
luedfunct
iondeﬁnedon(
a, b
)
is
said
to
be
convex
if
f
(
λx
+(1
−
λ
)
y
)
≤
λf
(
x
)+(1
−
λ
)
f
(
y
)
whenever
x, y