Math 825-826 Qualifying Exam Spring 2013
Work any six of the given problems, clearly marking the one problem you do not want
graded.
1. (20 points)
(a) Define limn→∞ sn = L and use your definition to prove limn→∞
√ √
√
n( n + 1 − n) = 12 .
(b) Give the definition of limx→−∞ f (x) = L and use your definition to prove that limx→−∞
x−1
x
= 1.
2. (20 points) Assume f : [a, b] → R is bounded. Define f is Riemann integrable on [a, b] in terms of
upper and lower Riemann sums. Prove that f is Riemann integrable on [a, b] if and only if given any
> 0, there is a partition P of [a, b] such that U (f, P ) − L(f, P ) < .
3. (20 points) Assume f : R → R is continuous. Let {δk }∞
k=1 be a decreasing sequence of real numbers
with limit 0. Define the sequence {hk }∞
by
k=1


f (k), x > k
hk (x) = f (x), |x| ≤ k


f (−k), x < −k.
Then define the sequence of the so called “integral means” {gk }∞
k=1 by for each x ∈ R gk (x) =
R x+δk
1
2δk x−δk hk (t) dt. Prove each of the following: each gk is continuous on R and |gk (x)| ≤ Mk :=
max{|f (x)| : |x| ≤ k}, on R, each gk is continuously differentiable and |gk0 (x)| ≤
limk→∞ gk (x) = f (x) uniformly on compact subsets of R. Finally, if
(
1, x ≥ 0
f (x) =
−1, x < 0
Mk
δk
on R. Prove that
find gk (x).
P
n −nx converges uniformly on [0, B] for any B > 0.
4. (20 points) Show that the infinite series ∞
n=0 x 2
Does this series converge uniformly on [0, ∞)? Prove your answer.
5. (20 points)
(a) Let (M, d) be a metric space, then if x0 ∈ M we define the open ball about x0 of radius r > 0
by Br (x0 ) = {x ∈ M : d(x, x0 ) < r}. Prove that Br (x0 ) is an open set in M . (Use the basic definition of an open set in Donsig’s book.)
(b) Let (X, ρ) be a complete metric space, assume1 A ⊂ X and F : X → X. Assume there is a
k > 0 such that ρ(F (a), F (b)) ≤ k · ρ(a, b) for all a, b ∈ X. Suppose further A ⊂ F (A). Provide,
with proof, a discription of A if k < 1. Can anything be said about the set A if k ≥ 1 ? (prove
or disprove your answer.)
6. (20 points) Prove Dini’s Theorem: Assume fn : [a, b] → R, is a decreasing sequence of continuous
functions with limn→∞ fn (x) = f (x) for each x ∈ [a, b], where the limit function f is continuous on
[a, b]. Show that limn→∞ fn (x) = f (x) uniformly of [a, b]. Give an example to show that we can not
replace [a, b] in Dini’s Theorem by (a, b). Give an example that shows that we can not delete the
assumption in Dini’s Theorem that the limit function is continuous.
Rb
7. (20 points) Prove that if f ∈ R(α) on [a, b] and α ∈ C 1 [a, b], then the Riemann integral a f (x)α0 (x)dx
exists and
Z
Z
b
b
f (x)dα(x) =
a
1
f (x)α0 (x)dx.
a
The statement should have included an additional assumption that A is bounded. Without it the question is too broad.