MAT 320 Introduction to Analysis Fall 1996
Review for Final Examination
References are to Reed, Fundamental Ideas of Analysis
pre-publication version, Wiley 1996
Questions will involve the following topics:
1. (A and B-tracks)
- Methods of proof: understand how to set up
and carry through a proof by induction [e.g. Proposition
1.4.3, Problems 7, 9 on p.23]. Be able to carry
out a proof by contradiction [e.g. Proposition 1.4.2].
Understand the words ``converse, contrapositive, counterexample.''
- Cardinality: understand how to show that 2 sets have the
same number of elements [Section 1.3, first paragraph,
Example 1, Proposition 1.3.3, etc.] Understand the proof
that the set of rational numbers is countable, and that
the set of reals between 0 and 1 is not [Theorems 1.3.4 and
1.3.5, Problem 8 on p.18].
- Convergence of sequences of real numbers: understand and be able to
apply the definition of ``the sequence {a_n} converges to
the limit L.'' Be able to calculate N from epsilon as in
[Problems 2 and 3 on page 31]. Be able to
prove that if the numbers L and L'
both are the limit of the sequence a_1, a_2, a_3, . . . ,
then L = L'.
- Be able to apply the limit theorems 2.2.3, 2.2.4, 2.2.5, 2.2.6
to calculate the limit of a sequence defined recursively [
Problem 1(a) on p.64, proof of Theorem 2.7.1, proof of
Theorem 2.7.2 (assuming the limit
exists, what is it?)]. Also [Problem 6 on page 37].
- Cauchy sequences: understand and be able to apply the
definition [Definition, p.43, Example 1]. Understand
the importance of the Completeness Axiom, and be able
to show that the rationals are not complete [discussion
on page 45, last paragraph].
- Every bounded monotone sequence of real numbers converges
to a limit [Theorem 2.4.3]. Understand how this theorem
is used in the proof of Theorem 2.7.1, for example, and in the
proof of the least upper bound property, Theorem 2.5.1.
- Least upper bounds: understand how their existence
is proved from the Completeness Axiom [Theorem 2.5.1];
know how to prove that least upper bounds are unique [
Problem 3 on p.52].
- The Bolzano-Weierstrass Theorem: understand what
a subsequence is; understand that there are bounded
sequences that do not converge to a limit, understand
the proof that a bounded sequence must have a convergent
subsequence [Theorem 2.6.2].
- Limit points: understand the definition (p.53),
and that limit points of a sequence are the same
as limits of subsequences [Proposition 2.6.1].
2. ( A-track: everything; B-track bold-face material.)
- Understand the definition of continuity in terms
of convergent sequences (p. 69); be able to use it to
prove that, for example, the function f(x) =
x^2 is continuous at x=1, and that the
function f(x)=0 if x< 1, =2
if x > =1 is not. (Examples 1 and 2 on pp. 70,71).
- Understand the ``delta-epsilon'' definition of
continuity and be able to prove that it is equivalent to
the ``convergent sequences'' definition (Theorem 3.1.3).
- Be able to prove: a continuous function on a closed
interval is bounded (Theorem 3.2.1). Understand why
a continuous function on a closed interval takes on its
maximum and minimum values, and all values in between
[Theorems 3.2.2 and 3.2.3].
- Understand the definition of uniform continuity
and be able to show, for example, that the function
f(x) = x^2 is not uniformly
continuous when considered as a function defined on the
whole number line. Be able to prove that a continuous
function on a closed interval is uniformly continuous (Theorem
3.2.5). Understand the definition of Lipschitz continuity
(Problem 7 p.83).
- Understand the definition of Riemann integrability
(p.86) for a bounded function on a finite interval. Be able
to show that the function which is 1 on rationals and 0
on irrationals (Example 1 p.99) is not Riemann integrable. Be able to
prove that a continuous function on a closed interval is Riemann
integrable (Theorem 3.3.1).
- Understand the definition of a Riemann sum (p.87)
and why for a continuous function on a
closed interval arbitrary Riemann sums converge to the
Riemann integral (Corollary 3.3.2). This justifies the use
of left-hand and right-hand sums in computations.
- Understand how the properties of the Riemann
integral given in Theorems 3.3.3, 3.3.4, 3.3.5,
Corollary 3.3.6, Theorem 3.3.7 follow from the definition
and Corollary 3.3.2. Understand how to use Theorem 3.3.5 (the
``triangle inequality for integrals") and be able to
prove it.
- Understand how the triangle inequalities (regular
and ``for integrals") are used in the estimate of the
error in a Riemann sum approximation in terms of a
bound on the derivative (Theorem 3.4.1). Be able to apply this
theorem to estimate how fine a partition is needed to
obtain a desired accuracy (Example 1 p.93).
-
Understand the definition of improper integrals in terms
of limits, as used in Examples 3,4,5 on pp.105,106.
Understand the convergence of the integral of sinx/x
(Example 6 p.106).
- Understand the definition of ``f is differentiable
at x,'' and be able to prove that this implies
continuity at x (Theorem 4.1.1). Be able to prove
the product rule (Theorem 4.1.2b).
- Be able to prove that the derivative vanishes at a
maximum, with the correct hypotheses (Theorem 4.2.1), as well
as Rolle's Theorem (Theorem 4.2.2) and the Mean Value Theorem
(Theorem 4.2.3).
- Be able to prove the version of the Fundamental Theorem
given as Theorem 4.2.5.
- Understand what the Taylor polynomials are and how
to compute them (p.134). Be able to quote and use Taylor's
Theorem estimating the error in approximating a
function by its nth Taylor polynomial (Theorem 4.3.1).
- Understand how Newton's method works. Understand
the necessity for the condition that the initial guess be
sufficiently close to the root in Theorem 4.4.1.
- Understand the definitions of pointwise convergence
and uniform convergence for a sequence of functions defined
on a set E. Understand why the sequence f_n=
x^n on [0,1] converges but does not converge
uniformly.
- Be able to prove that a uniform limit of continuous
functions is continuous.
- Understand the ``sup norm'' on the set of functions
defined on a set E, and that convergence in the
sup norm is the same as uniform convergence. Understand
what a Cauchy sequence of functions is (in the sup norm)
and be able to apply Theorem 5.3.2: A Cauchy sequence of
continuous functions converges uniformly to a limit which
is a continuous function.