Mathematics 741  Advanced Probability Theory I  Fall 2008  Zirbel
Analysis warmup.
These problems are representative of the analysis problems that arise in Mathematics
741.
They are in no particular order and are of varying difficulty levels, so try them all
rather than stopping when you find a problem you cannot solve. Attempt all problems. I
am interested in seeing partial solutions.
1.
Let
X
: Ω
→
R
.
Set
X
+
= max(0
, X
) and
X

= max(0
,

X
).
Show that
X
=
X
+

X

and that

X

=
X
+
+
X

. Do this by considering
X
(
ω
) for a given
ω
in Ω.
2.
The squeeze law.
Suppose that
a
n
≤
b
n
≤
c
n
for all
n
= 1
,
2
, . . .
, and that (
a
n
) and
(
c
n
) converge to the same finite limit
L
as
n
→ ∞
. Show that (
b
n
) converges to
L
as
well. Use an
ε

δ
argument, not lim sup.
3.
If (
a
n
) converges to a finite limit
a
as
n
→ ∞
, show that
lim
N
→∞
1
N
N
X
n
=1
a
n
=
a
.
4.
Let
T
⊂
R
, let
a
= sup(
T
), and suppose that
a <
∞
and that
a /
∈
T
. Show that there
exists a sequence (
a
n
)
⊂
T
for which
lim
n
→∞
a
n
=
a
.
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 Spring '10
 Zirbel
 Calculus, Set Theory, Ω, Basic concepts in set theory, lim sup

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