Math 3283W
Fall 2020
Time Limit: 120 minutes
This exam contains 8 numbered problems. Point values are in parentheses. No books, notes, or
electronic devices are allowed.
As on the writing quizzes, your work will be graded on the quality of your writing as well as on the
validity of the mathematics, but on this final exam there will be no separate writing score for any
of the problems. It is already included in the total points.
Do not use symbols for logical connectives and quantifiers unless directed to do so. That is, avoid
the use of the symbols ⇒, ⇔, ∧, ∨, ∼, ∀, ∃, and 3.
Axioms for Ordered Fields
A1. For all x, y ∈ F, x+ y ∈ R, and if x = w and y = z then
x + y = w + z.
A2. For all x, y ∈ F, x + y = y + x.
A3. For all x, y, z ∈ F, x + (y + z) = (x + y) + z.
A4. There is a unique element 0 ∈ F such that x+ 0 = x for
all x ∈ F.
A5. For each x ∈ F, there exists a unique element y such
that x + y = 0. (Often, we write y = −x.)
M1. For all x, y ∈ F, x · y ∈ R, and if x = w and y = z then
x · y = w · z.
M2. For all x, y ∈ F, x · y = y · x.
M3. For all x, y, z ∈ F, x · (y · z) = (x · y) · z.
M4. There is a unique element 1 6= 0 in F such that x ·1 = x
for all x ∈ F.
M5. For each x 6= 0 in F, there exists a unique element y
such that x · y = 1. (Often, we write y = 1/x.)
DL. For all x, y, z ∈ F, x · (y + z) = x · y + x · z.
O1. There is a relation < such that, for all x, y ∈ F, exactly
one relation holds: x = y, x < y or y < x.
O2. For all x, y, z ∈ F, if x < y and y < z then x < z.
O3. For all x, y, z ∈ F, if x < y then x + z < y + z.
O4. For all x, y, z ∈ F, if x < y and 0 < z then x · z < y · z.
may quote theorems from the book in your responses.
(a) Does there exist an surjective function from A = {1, 2} to B = {3, 4, 5}?
(b) Is the set of all irrationals in R countable?
(c) Does the set S =
{
n2 − 2n
3n− 7
∣∣∣n ≥ 2, n ∈ N} have a finite supremum?
2. (16 points) For each of the following subsets of R, list the set of boundary points, accumulation
points, and whether S is open, closed, or neither. (No justification is required in this question.)
(a)
⋃∞
n=1(0, n)
(b) (1, 4] ∩ (2, 6]
(c) N
(d) {(−1)n n+1n }
3. (8 points) Use the axioms in an ordered field (see cover page) to show that in any ordered field
F , given elements x and y in F , if x < y, then −y < −x.
4. (10 points) Prove that, for all natural numbers n,
1
1 · 2 +
1
2 · 3 +
1
3 · 4 + · · ·+
1
n(n + 1)
=
n
n + 1
5. (12 points) Find limx→1 f(x) for the function defined by
f(x) =
{
2x2−3x+1
x−1 for x 6= 1
5 for x = 1.
and prove your answer is correct using the definition of the limit of a function. (Do not rely on
any other results about limits from class.)
6. (12 points) Let f : A → B be any function and let D1, D2 be any non-empty subsets of B.
Proof or counterexample:
f−1(D1 ∪D2) = f−1(D1) ∪ f−1(D2)
7. (14 points) (a) State the definition of compactness (using the notion of open covers)
(b) State the Heine-Borel theorem
(c) Complete the following proof of the Bolzano-Weierstrass theorem (which states that every
bounded, infinite subset of R has an accumulation point):
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Proof by contradiction: Let S be a bounded set with infinitely many points, and such that S
has no accumulation points. Thus, S is closed, since it trivially contains all of its accumulation
points. So by the Heine-Borel theorem, S is compact.
Since S has no accumulation points, every point x ∈ S has a neighborhood Nx with Nx ∩ S =
{x}. Thus the set {Nx, x ∈ S} are an open cover. But...
8. (12 points) Find the limit of the sequence {sn} defined according to the relations s1 = 1 and
sn+1 =
1
5(sn + 7) for all n ≥ 1. Prove your answer is correct using the Monotone Convergence
Theorem.
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