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Math 301
First Exam
Thursday, Oct. 23, 2008
Try to give as much detail as possible in your answers. If you use a
theorem, state clearly which theorem you are using and show that the
hypotheses of that theorem apply.
1. a) State the definition of a Cauchy sequence of real numbers.
b) Suppose
z
∈ ℂ
, z
≠
1, and suppose m,n
∈
ℕ
with m < n. Write
(but do not prove!) a formula (a ratio of two terms)
for
z
m
+ z
m+1
+ ……….
.
+ z
n
.
c) Prove or disprove the following statement: If for all n, x
n
,
∈ ℝ
and
x
n
 x
n 
1
→
0, then the sequence { x
n
} is convergent.
d) Suppose {a
n
} is a sequence of complex numbers and
a
n+1
 a
n

≤
λ
n
 a
1
 a
0

,
where 0 < λ < 1. Prove that the sequence is a Cauchy sequence.
(Hint: You may assume as known that for any real number C,
C λ
n
→
0.)
2.
a) State the definition of lim sup for a sequence {a
n
} of real numbers.
b) State the Bolzano  Weierstrass Theorem.
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This note was uploaded on 10/07/2009 for the course MATH 301 at Yale.
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