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This package provides the full course class notes, homework answers, and exam questions for the course MATH521 (advanced calculus I) at UNC Chapel hill.

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• 11 artículos •
por mierli •
subido 2023

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Introduction to Advanced Calculus 1 (The Real Numbers) • Notas de lectura
Axiom of Completeness and Consequences • Notas de lectura
Consequences of Axiom of Completeness • Notas de lectura
Consequences from Axiom of Completeness Continued. • Notas de lectura
Sequences • Notas de lectura
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Compact Sets

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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Notas de lectura
• 4 páginas •
por mierli •
subido 2023

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Topology of R

Continue introducing the different parts to the topology of R, from the epsilon neighborhood of a, to open subsets of R, isolated points and limit points of A, then closed sets. Proof these different fundamentals of topology.

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Notas de lectura
• 2 páginas •
por mierli •
subido 2023

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University Of North Carolina - Chapel Hill • math 521
Advanced Calculus I (MATH521) • Por mierli

Sequences and Topology of R

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.

Book & Paket-Deal
Notas de lectura
• 4 páginas •
por mierli •
subido 2023

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Sequences

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.

Book & Paket-Deal
Notas de lectura
• 3 páginas •
por mierli •
subido 2023

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Limit of Sequence

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.

Book & Paket-Deal
Notas de lectura
• 2 páginas •
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subido 2023

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Algebraic Limit Theorem and Order Limit Theorem

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.

Book & Paket-Deal
Notas de lectura
• 3 páginas •
por mierli •
subido 2023

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Sequences

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.

Book & Paket-Deal
Notas de lectura
• 2 páginas •
por mierli •
subido 2023

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Consequences from Axiom of Completeness Continued.

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.

Book & Paket-Deal
Notas de lectura
• 3 páginas •
por mierli •
subido 2023

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Consequences of Axiom of Completeness

Explain the proof for the nested interval property, and also the proofs for the Archimedean Property, which is split into two parts.

Book & Paket-Deal
Notas de lectura
• 1 páginas •
por mierli •
subido 2023