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Lec 1 |
(10 Sept) |
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1.1
construction and the origin of numbers, natural
number as a system for counting, addition,
integers from the inverse of addition,
multiplication, the rationals from the inverse
of multiplication, the reals from geometry and
limits |
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Lec 2 |
(11 Sept) |
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1.2
characterization of the reals using axioms, the
axioms of a field (A1 -A9) |
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Lec 3 |
(15 Sept) |
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1.2
the axioms of an
ordered field (A1 - A13) (A14 is redundant) |
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Lec 4 |
(16 Sept) |
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1.3
identification of the natural numbers as the
minimal inductive subset of the reals, the
induction principle, proving properties of the
natural numbers using the induction principle |
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Lec 5 |
(17 Sept) |
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1.3 & 1.4
more on the use of the induction principle,
bounded subsets, supremeum and infimum |
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Lec 6 |
(18 Sept) |
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1.4
Dedekind completeness, the completeness axiom |
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Lec 7 |
(29 Sept) |
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1.4 more
on Dedekind completeness, the reals as the
unique Dedekind complete orderd field which
satisfies the Archimedean property |
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Lec 8 |
(30 Sept) |
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2.1-3
elementary real functions and the
epsilon-delta definition of limit |
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Lec 9 |
(01 Oct) |
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2.3
more on the intuition and the
epsilon-delta definition |
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Lec 10 |
(02 Oct) |
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2.4
basic properties of limits |
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Lec 11 |
(06 Oct) |
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2.4
more properties of limits: as a linear operator,
preserving products etc |
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Lec 12 |
(07 Oct) |
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2.4
more properties of limits:
preserving order, the squeeze rule, how to
convert intuitive ideas about limits into
rigorous epsilon-delta statements |
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Lec 13 |
(08 Oct) |
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2.5
continuity of a function, examples of
continuous and nowhere continuous functions |
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Lec 14 |
(09 Oct) |
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2.5
the
composition of continuous functions is
continuous |
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Lec 15 |
(13 Oct) |
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3.1, 2
limit point
(accumulation point) of a subset, definition of
open and closed subset of reals, motivation from
intervals, sequence definition of limit point,
examples, the construction of the Cantor set |
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Lec 16 |
(14 Oct) |
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3.2
the
Bolzano-Weierstrass Theorem, the theorem cannot
be strengthened by requiring only bounded from
above |
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Lec 17 |
(15Oct) |
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3.2 closed sets are preserved under
finite union and arbitrary intersection, similar
dual result for open sets, example showing an
infinite intersection of open sets needs not be
open |
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Lec 18 |
(16 Oct) |
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3.2 closure of a set, characterization of
openness using interior points |
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Lec 19 |
(20 Oct) |
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3.3
compact sets, the
intersection of a decreasing chain of nonempty
compact sets is a nonempty compact set |
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Lec 20 |
(21 Oct) |
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4.1,2
continuity of a
function and the preimage of an open set
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Lec 21 |
(22 Oct) |
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4.2
preservation of
compactness by continuous functions |
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Lec 22 |
(23 Oct) |
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4.2
preservation of
connectedness by continuous functions, i.e. the
Intermediate Value Theorem |
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Lec 23 |
(27 Oct) |
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4.2
the intermediate Value Theorem, any odd
degree polynomial over the reals has a
real root, real closed fields |
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Test II: 4:00 - 5:00 pm
27 October 2008 (covers Chapter 1 - 3 ) |
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Lec 24 |
(28 Oct) |
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Lec 25 |
(29Oct) |
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4.3
definition of differentiability,
4.4
the Mean Value Theorem |
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Lec 26 |
(30 Oct) |
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4.4
application of the MVT and the L'Hôpital's
Rule revision |
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