Mathematics 250
Fall 2008
Proof Practice # 1
MM, Exercise 2.5 of Chapter 2: Show that a closed ball
B
(
a
,r
) is indeed a closed set.
Argument 1
: We ﬁrst argue directly in terms of the deﬁnition of closed. A set
X
is said to be closed
if every sequence
{
x
n
}
∞
n
=1
of points in
X
, that converges in
R
k
, has its limit in
X
.
So, let’s check this for
X
=
B
(
~
a
,r
). Let
{
x
n
}
∞
n
=1
be an arbitrary convergent sequence in
B
(
~
a
,r
), and
let the limit of
{
x
n
}
∞
n
=1
be
L
. We want to show that
L
belongs to
B
(
~
a
,r
); that is, we want to show that

~
a

L
 ≤
r
. This will imply that
B
(
~
a
,r
) is closed.
Since
{
x
n
}
∞
n
=1
converges to
L
, we can ﬁnd points
x
k
that are arbitrarily close to
L
. That is, for every
± >
0, we can ﬁnd a
k
such that

x
k

L

< ±
. Also, since
x
k
is in
B
(
~
a
,r
), we know that

~
a

x
k
 ≤
r
.
Using these facts and the Triangle Inequality, we see that

~
a

L
 ≤ 
~
a

x
k

+

x
k

L
 ≤
r
+

x
k
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 Fall '06
 RogerHowe
 Math, #, Metric space, 1 mm, Topological space, RK, L

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