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Solutions to
131A Midterm 1, Spring 09.
Note: Proofs must be rigorous. Drawings will not be credited, though allowed.
Q1.
Let
A
and
B
be two subsets of
R
, both bounded below. Let
S
=
A
+
B
, the
set of elements of the form
a
+
b
with
a
∈
A
and
b
∈
B
.
a). Show that
S
is bounded below.
b). Show that
inf
(
S
) =
inf
(
A
) +
inf
(
B
).
Proof.
It is dual to the proof of Exercise 4.14 a) which is online (the midterm 1 review).
Go read it if you haven’t.
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View Full DocumentQ2.
Let
s
n
and
t
n
be two convergent sequences. Suppose
s
n
≤
t
n
for all but ﬁnitely
many
n
. Show that
lim
(
s
n
)
≤
lim
(
t
n
).
Proof.
Method of contradiction.
Put
lim
(
s
n
) =
s
and
lim
(
t
n
) =
t
for the sake of simplicity.
Suppose the opposite, i.e.,
s > t
. (Imagine a picture that on the
x
axis,
s
is to the
right of
t
.) Let
d
=
s

t
be the distance, and put
±
=
d
3
.
WLOG (Without loss of generality), we may assume
s
n
≤
t
n
is true for all
n
. See
the remark in the very end of this ﬁle (i.e., after
Q6
).
For this particular
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 Spring '09
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