Lecture Notes
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Notes
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Video
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Notes from Aug 24:
We discussed the syllabus, statements translated between English and Math, the triangle inequality, negating statements.
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Video from Aug 24
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Notes from Aug 26:
We made proofs and counterexamples regarding rational numbers and set theory.
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Not recorded (Sorry!)
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Notes from Aug 31:
We discussed proof examples, countability, sups, infs, axiom of completeness.
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Video from Aug 31
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Notes from Sep 2:
We discussed the Archimedian Principle, the Sup Lemma, the Density of Q in R, and the Nested Interval
Property of R. We found sups and infs of sets and used the lemmas to prove our results.
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Video from Sep 2
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Notes from Sep 7:
We discussed the definition of the limit of a sequence.
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Video from Sep 7
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Notes from Sep 9:
We disproved and proved that limits of sequences exist and stated the algebraic limit theorem.
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Video from Sep 9
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Notes from Sep 14:
We proved the square root part of the Algebraic Limit Theorem and also stated the Order
Limit Theorem. We considered some questions and proved our answers rigorously.
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Video from Sep 14
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Notes from Sep 16:
We discussed the squeeze theorem and monotone convergence theorem.
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Video from Sep 16
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Notes from Sep 21:
We discussed applications of the MCT, the fact that every convergent sequence is
bounded, and understanding series as sequences. We proved facts about the
geometric series.
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Video from Sep 21
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Notes from Sep 23:
We showed that the harmonic series diverges and proved that the terms of a convergent
series must converge to zero. We also computed some limits using the MCT.
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Video from Sep 23
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Notes from Sep 30:
We discussed subsequences and the Bolzano-Weierstrass Theorem. We also
proved a result about lim inf and lim sup.
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Video from Sep 30
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Notes from Oct 5:
We discussed the Cauchy Criterion and answered some True/False questions.
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Video from Oct 5
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Notes from Oct 7:
We discussed lots of facts concerning series, including the nth term lemma, eventual
convergence, the comparison test, the Cauchy Condensation Test, the p-series test,
the algebraic series theorem, the absolute convergence theorem, the alternating series test,
and the Riemann Rearrangement Theorem.
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Video from Oct 7
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Notes from Oct 12:
We discussed examples of sequences and series, the Riemann rearrangement theorem, and the ratio test.
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Video from Oct 12
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Notes from Oct 19:
We discussed open and closed sets.
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Video from Oct 19
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Notes from Oct 12:
We discussed examples of sequences and series, the Riemann rearrangement theorem, and the ratio test.
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Video from Oct 12
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Notes from Oct 21:
We discussed open and closed sets, closures, limit points, compact sets, and the Cantor set.
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Video from Oct 21
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Notes from Oct 26:
We discussed compact sets and the topological definition of compact.
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Video from Oct 26
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Notes from Oct 28:
We discussed connected sets in the set of real numbers. We also defined functional
limits and proved a few limits exist from the definition. We also stated the algebraic
and order limit theorems for functional limits.
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Video from Oct 28
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Notes from Nov 2:
We discussed some exercises and questions on topology and series.
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Video from Nov 2
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Notes from Nov 9:
We discussed the squeeze theorem for functional limits, definition of continuous,
algebraic continuity theorem, and compositions of continuous functions.
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Video from Nov 9
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Notes from Nov 11:
We discussed how to prove a functional limit does not exist. We also proved that the
continuous image of a compact set is compact, and as a corollary obtained the
extreme value theorem. We learned the definition of uniformly continuous, and we stated
the theorem that a continuous function on a compact set is automatically uniformly continuous.
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Video from Nov 11
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Notes from Nov 16:
We discussed uniform continuity, the continuous image of a connected set,
the intermediate value theorem, and the intermediate value property.
We defined differentiability, proved that differentiable implies continuous,
and we stated and proved part of the algebraic derivative theorem and chain rule.
We also stated the Darboux theorem, and we proved the interior extremum theorem
and Rolle's Theorem.
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Video from Nov 16
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Notes from Nov 18
Graph of Weierstrass function
We discussed the mean value theorem and consequences, and we defined Riemann integrability.
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Video from Nov 18
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Notes from Nov 30
We did an example of proving that a function is Riemann integrable from the definition.
After that, we proved that every continuous function is Riemann integrable, and we
showed that the Dirichlet function is not Riemann integrable on any interval [a,b], with a<b.
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Video from Nov 30
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Notes from Dec 2
We proved increasing functions are Riemann integrable. We discussed properties
such as linearity and monotonicity of the Riemann integral, and we proved the Fundamental
Theorem of Calculus. We also stated the Second Fundamental Theorem of Calculus and
the Riemann-Lebesgue Criterion.
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Video from Dec 2
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Notes from Dec 7
We discussed exercises that involved Riemann integration and the first and second fundamental theorems of calculus.
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Video from Dec 7
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