Takehome Midterm
Math 361, Spring 2012
These problems must be written up in LATEX, and are due on Sunday, March 25.
Rules: This is a midterm exam, not a homework assignment. You must solve the problems
entirely on your own, and you should not discuss the problems with any other students in
the class, or with anyone on the internet. When working on the problems, you may consult
the textbook, your class notes, your old homework assignments, and the materials on the
class webpage; you should not consult any other sources, including internet websites.
∞
∞
1. Let an n=1 and bn n=1 be Cauchy sequences in R.
∞
(a) Using only the definition of Cauchy sequences, prove that an +bn n=1 is a Cauchy
sequence.
∞
(b) Using only the definition of Cauchy sequences, prove that an bn n=1 is a Cauchy
sequence.
2. Let A ⊆ R be a nonempty set, let f : R → R be a continuous function, and suppose
that A and f (A) are bounded. Prove that f (lub A) ≤ lub f (A).
3. Let A ⊆ R be a set, and let f, g : A → R be continuous functions. Suppose that
for every x ∈ A and every ε > 0, there exists a y ∈ A so that |x − y| < ε and
|f (y) − g(y)| < ε. Prove that f (x) = g(x) for all x ∈ A.
4. Let f : (0, ∞) → R be a continuous function, and suppose that f (1/n) = (−1)n for all
∞
n ∈ N. Prove that there exists a sequence an n=1 in (0, ∞) such that f (an ) = 0 for
all n ∈ N and lim an = 0.
n→∞
∞
5. Let an n=1 be a convergent sequence of real numbers, and let φ : N → N be a bijection.
∞
Prove that the sequence aφ(n) n=1 converges, with lim aφ(n) = lim an .
n→∞
n→∞
6. Let f : R → R be a uniformly continuous function, and suppose that f (n) = 0 for all
n ∈ Z. Prove that f is bounded.