Subsequences: bounded sequences and Bolzano-Weierstrass theorem

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Convergence and Cauchy Sequences

Covers convergence, Cauchy sequences, and limit superior and limit inferior of bounded sequences.

Limit Superior and Limit Inferior

Explores limsup, liminf, Bolzano-Weierstrass theorem, accumulation points, and bounded sequences.

Uniform Integrability and Convergence

Explores uniform integrability, convergence theorems, and the importance of bounded sequences in understanding the convergence of random variables.

Sequences and Series: Defined and Convergent

Covers the definition of sequences, subsequences, limits, and series, including Cauchy sequences and convergence.

Convergence and Cauchy Sequences

Explores convergence and Cauchy sequences, including the Bolzano-Weierstrass theorem and the properties of convergent sequences.

Multivariable Integral Calculus

Covers multivariable integral calculus, including rectangular cuboids, subdivisions, Douboux sums, Fubini's Theorem, and integration over bounded sets.

Calculus of Variations: Principles and Applications

Explores the principles and applications of calculus of variations, focusing on uniform boundedness and equi-integrability of Carathéodory integrands.

Quadratic Penalty Method: Finer Analysis

Covers the quadratic penalty method and augmented Lagrangian, including the setup and convergence of sequences.

Demonstration of Bolzano-Weierstrass

Covers the demonstration of the Bolzano-Weierstrass theorem for real numbers sequences.

Subsequences and Bolzano-Weierstrass Theorem

Covers the squeeze theorem, monotone sequences, subsequences, and Bolzano-Weierstrass theorem, emphasizing the importance of peaks in sequences.