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
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 Fall '08
 ALBERTERKIP
 Math, Topology, Trigraph, 2 K, Metric space, Topological space, Archimedean

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