MAT 320 Fall 2007
Review for Midterm 2
•
Theorem: be able to apply
•
Theorem
: and know what goes into the proof
•
Theorem
: and be able to prove.
§
3.3
Monotone Convergence Theorem
. The least upper bound property
is crucial. Understand Example 3.3.3(b).
§
3.4 Theorem 3.4.4.
Monotone Subsequence Theorem, Bolzano Weier
strass Theorem (first proof)
.
§
3.5 Know definition of Cauchy sequence.
Cauchy Convergence Crite
rion
.
Proof uses
Lemma
: a Cauchy sequence is bounded; then Bolzano
Weierstrass to produce a candidate limit; then additional

δ
argument to
show that limit works.
Know definition of contractive sequence.
A contractive sequence is a
Cauchy sequence
. Proof is straightforward once you use
a
k
+
a
k
+1
+
· · ·
+
a
k
+
‘
=
a
k
(1

a
‘
+1
)
/
(1

a
).
§
3.6 Know definition of properly divergent sequence.
§
4.1 Know definition of cluster point and of limit of a function at a cluster
point.
Theorem 4.1.5
(uniqueness of limit): basic and paradigmatic

δ
argument.
Sequential Criterion for Limits (Theorem 4.1.8)
.
Divergence
Criteria (4.1.9).
§
4.2
Theorem 4.2.2
(
f
has a limit at
c
implies
f
is bounded on a neigh
borhood of
c
).
Nice

δ
argument.
Theorem 4.2.3
on limits of sums,
products, quotients.
Theorem 4.2.6
on
≤
inequalities persisting to limits.
Theorem 4.2.9
(lim
x
→
c
f
(
x
)
>
0 implies that
c
has a
δ
neighborhood on
which
f
(
x
)
>
0): useful theorem and illustrative proof.
§
4.3 Not necessary to review for test. Check exercises.
§
5.1 Know definition of “
f
continuous at
c
” in

δ
terms. Understand Re
mark after Theorem 5.1.2.
Sequential Criterion for Continuity
. Know
5.1.6 Examples (g) and (h).
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 Fall '10
 Mark
 Calculus, Continuous function, Bolzano Weierstrass Theorem, Interior Extremum Theorem

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