Analysis Qualifying Exam, June 7, 2014
This exam has five questions. Please answer each question as completely as possible.
Unsupported work will receive no credit, and partially completed work may receive partial
credit. Each question is worth 5 points, for a total of 25 points. Good luck to you all!
Note: Well-known theorems may be used without proof, unless otherwise stated.
1. Let  be a sequence of real numbers satisfying
|+1 −  | ≤ − for all  ≥ 1
where   1 is a constant. Prove lim  exists.
→∞
2. (a) Assume
∞
P
=1
Does
∞
P
 is convergent and { } is a bounded sequence.
  converge? Prove or provide a counterexample.
=1
(b) Assume
∞
P
=1
Does
∞
P
| | is convergent and { } is a bounded sequence.
  converge? Prove or provide a counterexample.
=1
3. Consider the series
∞
X
(−1) 

+
=1
(a) Prove this series is convergent pointwise for all  ∈ [0 1]
(b) Prove this series is uniformly convergent for all  ∈ [0 ] where  is a constant
and 0    1
(c) Is this series uniformly convergent for all  ∈ [0 1]? Prove your answer.
4. (a) Define what it means for a bounded function on [ ] to be Riemann integrable.
(b) Use your definition from (a) to prove the following function  () is Riemann
integrable on [0 2]:
(

0≤≤1
 () =
2 + 1 1   ≤ 2
5. Let  be the following set:
½
¾ ½
¾ ½
¾
1
1
1
1
=
: ≥1 ∪ − : ≥1 ∪
−
:   ≥ 1


 
(a) Find all limit points of 
(b) Prove  is a compact set using the open cover definition.
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