MAT 320 Introduction to Analysis Fall 1996
Review for Midterm 1
References are to Reed, Fundamental Ideas of Analysis
pre-publication version, Wiley 1996
There will be four questions, each worth 25 points.
One question will be: Prove that if the numbers L and L'
both are the limit of the sequence a_1, a_2, a_3, ...
then L = L'.
The other questions will involve the following topics:
- 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 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].