Chapter 0
Prepreliminaries
0.1
Course Overview
The study of the real numbers,
R
, and functions of a real var,
f
(
x
) =
y
where
x, y
real.
Given
f
:
R
→
R
which describes some system, how to study
f
?
•
Need rigourous vocab for properties of
f
(definitions)
•
Need to see when some properties imply others (theorems)
Result: can make inferences about the system.
Motivation for analysis: limits, the heart & soul of calculus.
Limits provide a rigourous basis for ideas like sequences, series, continuity, derivatives,
integrals.
More adv:
model an arbitrary function as a limit of a sequence of “nice”
functions (polys, trigs) or as a sum of “nice” functions (Fourier, wavelets).
All of this
requires understanding limits of numbers.
Outline:
1. Logic: not, and, or, implication; rules of inference
2. Sets: elements, intersection, union, containment; special sets
3. The real numbers: algebraic properties (+
,
×
), order properties (
<
), completeness
properties
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8
Math 413
Prepreliminaries
4. Sequences: types of, convergence, basic results (arithmetic, etc), subsequences, con
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 Fall '11
 Wong
 Calculus, Logic, Real Numbers, Mathematical analysis

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