Review sheet for Midterm 1, MAT 319, Foundations of Analysis, Spring
2013
1. Mathematical induction. You need to be able to clearly state the princi
ple of mathematical induction and do a inductionbased proof. Sample
problems: 1.2, 1.5, 1.6, 1.9.
2. Fields, properties of
Q
and
R
.
I will not give any problems directly
on this. Don’t bother memorizing the axioms in Paragraph 3, this is
something ”you already know”. However, make sure you are confident
with inequalities, absolute value, and the triangle inequality.
3. The completeness axiom.
Definitions you need to know: maximum,
minimum, upper bound, lower bound, least upper bound, greatest lower
bound (make sure you know exactly what the difference between these
is). You need to be able to state the Completeness Axiom (4.4), the
Archimedean axiom (4.6) and the denseness of
Q
(4.7). Sample prob
lems: 4.14.4 (try to do these in your head), 4.5, 4.6, 4.9, 4.15.
4. Infinity. You need to know the conventions about using the symbols
+
∞
and
∞
(page 27).
5. Dedekind cuts. You need to know the definitions (i)(iii) in Paragraph
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 Spring '08
 Staff
 Mathematical Induction, Order theory, completeness axiom

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