Final Exam - Kalamazoo College

Math 320, Real Analysis I
Final Exam
Instructions: This take-home final exam is due by 12:00 p.m. (noon) on Wednesday, June 6.
You should answer each question with a well-written, coherent, clear, and mathematically precise
short essay, just as you have done in your weekly assignments. Both the correctness of the
mathematics and the quality of your exposition will influence your grade. Don’t turn in first draft
material!
While you work on the test, you may consult your textbook, other books, notes, and all handouts
you received in class. You may also come and talk to me. No other sources of information are
allowed; in particular, you should work by yourself without the help of others, and you shouldn’t
search the internet to find solutions. On your examination paper, please write
I have followed the instructions and the Kalamazoo College Honor Code for this test.
and sign your name.
Good luck!
Part I: Show me one!
Give an example of each object described below. Explain why your example does what you claim
it does. These questions are worth 5 points each, for a total of 25 points.
1. A Cauchy sequence in Q that does not converge in Q.
2. A connected subset A ⊆ R that is neither open nor closed.
3. A sequence of functions that converges pointwise on A ⊆ R, but not uniformly on A.
4. A bounded sequence of functions that has no uniformly convergent subsequence.
5. A series of functions that converges on an interval (a, b), but does not converge on R.
Part II: Properties of the Uniform Norm
This section is worth a total of 25 points.
Suppose f : A → R is a bounded function. Define the uniform norm of f on A by
||f|| A = sup{ |f(x)| : x ∈ A }.
Show that the uniform norm satisfies the following conditions:
1. ||f|| A ≥ 0 for all x ∈ A, and ||f|| A = 0 if and only if f(x) = 0 for all x ∈ A.
2. ||λf|| A = |λ| · ||f|| A for all λ ∈ R.
3. Whenever f, g : A → R are both bounded functions, show that
||f + g|| A ≤ ||f|| A + ||g|| A .
4. Whenever f, g : A → R are both bounded functions, show that
||f g|| A ≤ ||f|| A · ||g|| A .

Math 320, Real Analysis I
Final Exam
Part III: Taylor Series and Lagrange’s Remainder Theorem
These questions are worth 10 points each, for a total of 30 points.
Let f be infinitely differentiable on the interval (−R, R). Define a n = f (n) (0)
n!
let
S N (x) = a 0 + a 1 x + a 2 x 2 + · · · + a N x N .
for each n, and
The polynomial S N (x) is a partial sum of the Taylor series expansion for the function f(x). Define
E N (x) = f(x) − S N (x)
which represents the error between f and the partial sum S N . Clearly proving S N (x) → f(x) is
equivalent to showing E N (x) → 0.
1. Explain why the error function E N (x) = f(x) − S N (x) satisfies
for all n = 0, 1, 2, . . . , N.
E (n)
N (0) = 0
2. Assume x > 0 and consider the functions E N (x) and x N+1 on the interval [0, x]. Show that
there exists a point x 1 ∈ (0, x) such that
E N (x)
x N+1 = E′ N (x 1)
(N + 1)x N 1
3. Prove Lagrange’s Remainder Theorem, which states:
Let f be infinitely differentiable on (−R, R), define a n = f (n) (0)/n!, and let
S N (x) = a 0 + a 1 x + a 2 x 2 + · · · + a N x N .
Given x ≠ 0, there exists a point c satisfying |c| < |x| where the error function
E N (x) = f(x) − S N (x) satisfies
.
E N (x) = f (N+1) (c)
(N + 1)!
x N+1 .
Part IV: Arzelà-Ascoli Theorem
Question 1 is worth 5 points while the remaining questions are 10 points each, so this section is
worth a total of 45 points.
Recall the Bolzano-Weierstrass Theorem (Theorem 2.5.5) states that every bounded sequence
of real numbers has a convergent subsequence. An analogous statement for bounded sequences
of functions is not true in general. (See Part I, Problem 4.) Under stronger hypotheses,
however, we can derive a Bolzano-Weierstrass Theorem for sequences of functions. First, we make
the following definition:
A sequence of functions (f n ) defined on a set E ⊆ R is called equicontinuous if for every
ε > 0 there exists a δ > 0 such that
for all n ∈ N and |x − y| < δ in E.
|f n (x) − f n (y)| < ε

Math 320, Real Analysis I
Final Exam
With this definition now in hand, answer the following question.
1. What is the difference between saying that a sequence of functions (f n ) is equicontinuous and
just asserting that each f n in the sequence is individually uniformly continuous?
For each n ∈ N, let f n be a function defined on [0, 1]. If (f n ) is bounded on [0, 1] — that
is, there exists an M > 0 such that |f n (x)| ≤ M for all n ∈ N and all x ∈ [0, 1] — and if
the collection of functions (f n ) is equicontinuous, follow these steps to show that (f n ) contains a
uniformly convergent subsequence.
2. By Exercise 6.2.14, there exists a subsequence (f nk ) that converges at every rational point in
[0, 1]. To simplify notation, set g k = f nk .
IGNORE THE REST OF THIS PROBLEM! It is a typo, as it is only asking you
to do problems 3, 4 and 5 below. I’ll give each of you the 10 points from this
problem “for free” when grading the exam.
Show that (g k ) converges uniformly on all of [0, 1].
3. Let ε > 0. By equicontinuity, there exists a δ > 0 such that
|g k (x) − g k (y)| < ε 3
for all |x − y| < δ and k ∈ N. Using this δ, show that there is a finite collection of rational
points r 1 , r 2 , . . . , r m ∈ [0, 1] with the property that the union of the neighborhoods V δ (r i )
contains [0, 1].
4. Explain why there must exist an N ∈ N such that
|g s (r i ) − g t (r i )| < ε 3
for all s, t ≥ N and r i in the finite subset of [0, 1] described above.
Why does having the set {r 1 , r 2 , . . . , r m } be finite matter?
5. Finish the argument by showing that, for an arbitrary x ∈ [0, 1],
|g s (x) − g t (x)| < ε
for all s, t ≥ N.