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Compact Sets
- Class notes • 4 pages • 2023
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Introduce and provide proof for compact sets, that is a set that is both bounded and closed, hence the Heine-Borel theorem. Then introduce the perfect set, the connected/disconnected sets, and lastly the canter set, write proofs for all three.
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Sequences and Topology of R
- Class notes • 4 pages • 2023
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Finish the lecture on sequences and the convergence of Cauchy. Then introduce the topology of R and in it the epsilon neighborhood of a in R. Define open sets, limit points, isolated points, and closed subsets.
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Sequences
- Class notes • 3 pages • 2023
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Introduce sequences and its epsilon proof. Then explain sequences with the algebraic limit theorem and the order limit theorem. Write proof for the theorem that a sequence converges if and only if that sequence is a Cauchy sequence.
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Consequences from Axiom of Completeness Continued.
- Class notes • 3 pages • 2023
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Explain and prove more consequences from the axiom of completeness. First, the density of rational numbers in real numbers using Archimedean principle 1 and axiom of completeness. Then prove the existence of the square root of 2 with contradiction. Lastly, the sequences argument with the algebraic limit theorem and order limit theorem.
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Axiom of Completeness and Consequences
- Class notes • 3 pages • 2023
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Introduce Axiom of Completeness, the bed rock of advanced calculus, where every nonempty bounded above subset of R has a lower upper bound (supremum), and every nonempty bounded below subset of R has a greatest lower bound (infimum). Introduce the consequences of the Axiom of Completeness, with the Nested Interval Property and the Archimedean Property.
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Algebraic Limit Theorem and Order Limit Theorem
- Class notes • 3 pages • 2023
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Explain and provide proof for the algebraic limit theorem and the order limit theorem. Use proof by contradiction for the assumption and conclusion of the two sequences.
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Limit of Sequence
- Class notes • 2 pages • 2023
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Give an epsilon proof for the limit of a sequence, where in more detail the partial sum of a series and its convergence is explained and tested with the p-series test. Then, introduce the Cauchy sequence and the monotone convergence theorem.
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Sequences
- Class notes • 2 pages • 2023
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Introduce sequences and its epsilon proof. Define the convergence of an to a and how to prove it with a backwards proof by first identifying the candidate for the limit and prove that that is the candidate, then prove convergence. Then introduce some theorems about sequences, namely the algebraic limit theorem and the order limit theorem.
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Consequences of Axiom of Completeness
- Class notes • 1 pages • 2023
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Explain the proof for the nested interval property, and also the proofs for the Archimedean Property, which is split into two parts.
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