This preview shows pages 1–2. Sign up to view the full content.
Math 301 Midterm. Solutions
1. (40 points) In your answers you should refer to de&nitions or theo
rems/lemmas.
(a) Is there a bounded increasing sequence in
R
which is not Cauchy?
If yes give an example, otherwise prove that there is no such ex
ample.
(b) Consider the decimal expansion of
= 3
:
141 592 654
:::
and form
a sequence
(
x
n
)
1;
then
x
1
=
1
10
:
Take the next two digits
41;
then
x
2
=
41
100
;
the next
3 digits are
592
so
x
3
=
592
1000
and so on (then
x
n
=
abc:::z
10
n
where
abc:::z
are the "next"
n
digits). Does
(
x
n
)
have any convergent
subsequence? Prove your answer.
(c) Let
K
be some closed set in the metric space
(
M;d
)
and let
a
be
some element of
M
with
a
62
K:
Prove that there is some
± >
0
so
that
D
(
a;±
)
\
K
=
²:
(d) Prove that
\
1
n
=1
&
0
;
1
n
±
=
²
. Which property of
R
are you using?
Solution 1
Each part is 10 points
(a) No, because a bounded increasing sequence in
R
is convergent by
the Completeness axiom and any convergent sequence is Cauchy.
(b) The sequence is bounded, all terms are in the interval
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.
 Fall '08
 ALBERTERKIP
 Math

Click to edit the document details