Real Analysis Qualifying Exam, September 16, 2015
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 f : R → R be a continuous function such that
lim f (x) = lim f (x) = +∞.
x→+∞
x→−∞
Prove that f attains an absolute minimum value on R. In other words, prove that there exists a
real number c such that f (c) ≤ f (x) for all x ∈ R.
2. Let {xn } be a sequence of real numbers satisfying
|xn+1 − xn | ≤ C|xn − xn−1 |,
for all n ≥ 1, where 0 < C < 1 is a constant. Prove that {xn } converges.
3. Prove that for all real numbers x and y,
| cos2 (x) − cos2 (y)| ≤ |x − y|.
4. Consider
f (x) =
∞
X
1
sin(2k x).
2k
k=0
(a) Show that f is continuous on R.
(b) Show that f is not differentiable at x = 0. (Hint: Consider the sequence {xn } =
π
2n
.)
5. (a) State the definition for a real valued function f : [a, b] → R to be Riemann integrable on the
interval [a, b].
(b) Use the definition to prove that f (x) = ln(x) is Riemann integrable on the interval [1, 2].
Note: If you choose to work with a definition of Riemann integrability different than that stated
in part (a), please provide the alternate definition.
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