Real Analysis Qualifying Exam
Date: September, 19 2009.
Duration: 2 Hours
Instructions: This exam consists of 5 questions. Each question is worth 5 points giving a grand total
of 25 points possible. Please present all of your work in a clear and concise manner and answer each
question as completely as possible. Unsupported work will receive no credit and partially completed work
may receive partial credit. Good luck!
1. Let A ⊆ R be a non-empty bounded subset of R.
(a) Prove that there exists a sequence of real numbers in A that converge to sup A.
(b) Let b ∈ R. Suppose that b is an upper bound for A and that there exists a sequence of real numbers
in A that converge to b. Prove that b = sup A.
2. (a) Let {ak } be a sequence of real numbers that converge to 0. Prove that
only if
∞
X
∞
X
ak is convergent if and
k=1
(a2k−1 + a2k ) is convergent.
k=1
(b) Provide an example of a sequence of real numbers {ak } such that the series
convergent but the series
∞
X
∞
X
(a2k−1 + a2k ) is
k=1
ak is divergent. Please justify your claim for full credit.
k=1
3. Let n ∈ N and let c0 , . . . , cn ∈ R. Let a ∈ R and suppose that
c0 +
cn−1
cn
c1
+ ··· +
+
= a.
2
n
n+1
Show that there exists a real number x ∈ R such that
c0 + c1 x + · · · + cn−1 xn−1 + cn xn = a.
Hint: Construct a function that is motivated by the given information.
4. Let {an } be a sequence of real numbers that converge to a ∈ R and let f : R → R be a uniformly
continuous function. For each n ∈ N, let fn : R → R be the function defined by fn (x) = f (x + an ) for
x ∈ R. Show that the sequence of functions {fn } converge uniformly to g on R where g(x) = f (x + a).
5. For each n ∈ N, let xn =
1
n
and define fn : [0, 1] → R by
(
0 if x ∈ {x1 , . . . , xn }
fn (x) =
.
1 if x ∈ [0, 1] \ {x1 , . . . , xn }
(a) Using the definition of Riemann integrability, show that for each n ∈ N, the function fn is Riemann
Z 1
integrable on [0, 1] with
fn (x) dx = 1.
0
(b) Determine the pointwise limit of the sequence of functions {fn } on [0, 1] and prove that this limit is
Riemann integrable on [0, 1].
1