Download Prerequisites and some problems

Prerequisites and some problems
I assume you’re familiar with the material of Math 300, 327, 328. Presumably you’ve
also had multivariable calculus and linear algebra. Here’s a list of some of the key topics
involved; if rusty be sure to review in advance!
1. Logic and proofs. Converse, biconditional, contrapositive. Negation, including negation of statements with multiple quantifiers ∀, ∃. Proof by contradiction. Proof by induction,
including strong induction.
2. Set theory and functions. Basic set theory: unions, intersections, cartesian products
etc. Functions f : X−→Y from one set to another (by the way, let’s call X the domain and Y
the codomain. Our text uses the term “range” for the codomain.) Images and inverse images
(inverse images are also known as “preimages”). Injections, surjections and bijections. Finite
sets and infinite sets. Countable and uncountable sets, e.g. Q is countable but R is not.
3. Rigorous definition of limits for real numbers, using epsilons and deltas in all their
glory, e.g. limits of sequences limn→∞ xn = x, limits of functions limx→a f (x) = A. In
particular, the rigorous definition of continuity of a function.
4. Basic point-set topology of the real line (you might not have called it topology, but
it is!): Least upper bound property, Bolzano-Weierstrass theorem, Cauchy sequences, limit
points (also known as accumulation points), open and closed subsets, etc.
Now, here are some problems to test your understanding of some of the background
material, and simultaneously review it. Many of these appear as theorems or exercises from
prerequisite texts, but it is absolutely essential to resist looking up solutions. Don’t do this
unless you’re really stuck! Reading a proof cold before you’ve given a problem any thought
is not likely to be productive. But if you’ve really pondered the problem before resorting
to outside help, your brain will be primed for enlightenment. In any case, don’t go into a
panic if you can’t do some of these problems. The important thing is to give it a serious
try before looking at someone else’s solution. To help you resist temptation, I won’t give
solutions here.
1. Let xn be a sequence of real numbers. Let f : R−→R be a function that is continuous
at a point a ∈ R. Using the rigorous δ definitions, show that if limn→∞ xn = a, then
limn−→∞ f (xn ) = f (a).
2. Recall that Rn is the set of n-tuples (x1 , ..., xn ) of real numbers (often thought of as
vectors). The (n − 1)-sphere S n−1 is the subset of vectors of unit length, i.e.
S n−1 = {(x1 , ..., xn ) ∈ Rn :
X
x2i = 1}.
For example, S 1 is the unit circle in the plane, and S 2 is the usual unit sphere in 3-space.
But don’t let the higher-dimensional cases intimidate you; when you’ve seen one you’ve seen
them all.
1
Let R+ denote the positive real numbers, and define f : R+ × S n−1 −→Rn − {(0, ..., 0)} by
f (t, (x1 , ..., xn )) = (tx1 , ..., txn ). Show that f is bijective.
3. A very important theorem (that should be familiar) states that every Cauchy sequence
of real numbers converges. In this problem you’ll prove this theorem with hints. If stuck,
look it up in your 327 textbook for instance—but only when you’re really stuck!
a. State the rigorous definition of “Cauchy sequence”.
b. Negate the statement of (a). This is a good example of “negating a statement with
multiple quantifiers”.
c. Now, first show every Cauchy sequence is bounded. Do this by contrapositive, making
use of (b): i.e. show that if the sequence isn’t bounded, then it isn’t Cauchy.
d. A famous theorem tells us that every bounded sequence has a convergent subsequence.
Name that theorem!
e. In view of parts c,d, every Cauchy sequence {xn } has a convergent subsequence {xnk },
converging let’s say to a. Show that in fact the original sequence xn also converges to a,
thereby gloriously completing the proof.
4. Show that the following two sets are countable. Do not use explicit bijections; make
your life easier by applying suitable theorems from §1.7 of our text.
a) Q, the rational numbers.
b) Let S denote the set of all sequences of rational numbers that are eventually zero. Here
a sequence x1 , x2 , ... is “eventually zero” if there exists N such that xi = 0 for all i > N ; in
other words, the sequence has only finitely many nonzero terms.
Well, you may not have the text yet, so here are some “suitable theorems” you might
apply:
• Every subset of a countable set is countable.
• If X is countable and f : X−→Y is surjective, then Y is countable.
• Any countable union of countable sets is countable. (This means “the union of a
countable number of countable sets”.)
• Any finite cartesian product of countable sets is countable. In other words, if X1 , ...Xn
are countable, then so is their cartesian product X1 × X2 ... × Xn .
5. The purpose of this problem is to practice proof by induction, but while we’re at it
we may as well prove something important, namely:
Theorem 1. Let f (x) = an xn + an−1 xn−1 + ... + a1 x + a0 be a polynomial with real
coefficients, with an 6= 0 (we say that f has degree n; note this excludes the identically zero
polynomial). Then f has at most n real roots. In particular, f has only finitely many real
roots.
a) Prove this by induction on n, using the division theorem for polynomials (explained
below).
2
The division theorem for polynomials is analogous to division with remainder for integers,
and reads as follows:
Theorem 2. Let f, g be polynomials with real coefficients, with g not the zero polynomial.
Then there are polynomials s and r such that f = sg + r and the degree of r is less than the
degree of g (we allow r to be the zero polynomial here).
b) Prove the division theorem by strong induction on the degree of f . (In other words,
consider a fixed polynomial g and prove the existence of s, r by induction on the degree n of
f .)
Suggestion: At the inductive step you’ll have a polynomial f of degree n, with inductive
hypothesis that the theorem is true for polynomials of degree < n. Let m be the degree of
g, and consider two cases: Case 1. n < m. This case is trivial; why? Case 2. n ≥ m. Show
there is a constant c such that the polynomial f − cxn−m g has degree < n, and therefore the
inductive hypothesis can be applied to it.
Remarks: 1. Both theorems are valid with complex coefficients too. In fact, if you’ve
seen a little abstract linear algebra and know the definition of a “field”, I’ll mention that
both theorems are valid with coefficients in any field.
2. Theorem 1 is an important fact of life in the world of algebra. It will come up in some
of our topological examples too.
3