MA361 Midterm Study Guide

The midterm examination will be held Thursday, October 29th.

The exam will consist of three sections:

A matching section with some terms and their definitions, also some theorems and their statements
Two proofs you have seen before. Two of the following four proofs will appear on the exam:
- Prove that every convergent sequence is bounded (Theorem 2.8)
- Prove the Comparison Theorem (Theorem 2.17)
- Prove the Monotone Convergence Theorem (Theorem 2.19)
- Prove that a Cauchy sequence in R converges (Theorem 2.29)
Two proofs you may or may not have seen before, chosen from a list of four possibilities.

Obviously due to time limitations the latter set of proofs will have to be relatively short ones.

The terms will be taken from the following list:

Term	Text Reference
one-to-one	Definition 1.29
onto
bijection
bounded above	Definition 1.10
supremum	Definition 1.10
bounded below	Definition 1.19
infimum
bounded
R,N,Z,Q	Section 1.2 p.7
image of a set under f	Definition 1.33
inverse image of a set under f	Definition 1.33
finite set	Definition 1.38
countable set
at most countable set
uncountable set
convergent sequence	Definition 2.1
subsequence	Definition 2.5
bounded (above/below) sequence	Definition 2.7
divergent sequence	Definition 2.14
increasing sequence	Definition 2.18
decreasing sequence
monotone sequence
Cauchy sequence	Definition 2.27
limsup	Definition 2.32
liminf	Definition 2.32
function limit (2-sided)	Definition 3.1
left hand function limit	Definition 3.12
right hand function limit	Definition 3.12
function limits involving infinity	Definition 3.15
continuity	Definition 3.19
uniform continuity	Definition 3.35

The theorems (on the matching section) will be taken from the following list:

Term	Text Reference
fundamental theorem of absolute values	Theorem 1.6
approximation property for suprema	Theorem 1.14
completeness axiom	Postulate 3 p.18
Archimedean principle	Theorem 1.16
reflection principle	Theorem 1.20
monotone property	Theorem 1.21
well-ordering principle	Theorem 1.22
DeMorgan's laws	Theorem 1.36
at most countable characterization	Lemma 1.40
Bolzano-Weierstrass theorem	Theorem 2.26
comparison theorem (functions)	Definition 3.10
sequential characterization of continuity	Theorem 3.21
preservation of Cauchy sequences	Lemma 3.38