Lectures related to Subsequences: bounded sequences and Bolzano-Weierstrass theorem

Sequences and Convergence: Understanding Mathematical Foundations

Covers the concepts of sequences, convergence, and boundedness in mathematics.

Convergence and Limits in Real Numbers

Explains convergence, limits, bounded sequences, and the Bolzano-Weierstrass theorem in real numbers.

Bolzano-Weierstrass: Advanced Analysis I

Explores the Bolzano-Weierstrass theorem on bounded sequences and convergent subsequences.

Subsequences and Bolzano-Weierstrass Theorem

Covers the proof of the Squeeze Theorem, Quotient Criteria, and the Bolzano-Weierstrass Theorem.

Bolzano-Bastra-Sterl: Applications and Uniform Continuity

Explores the Bolzano-Bastra-Sterl theorem and uniform continuity in sequences.

Convergent Sequences: Definitions and Illustrations

Explains convergent sequences, bounded sequences, subsequences, and compact sets with illustrations and proofs.

Banach Spaces: Reflexivity and Convergence

Explores Banach spaces, emphasizing reflexivity and sequence convergence in a rigorous mathematical framework.

The Bolzano-Weierstrass Theorem

Explains the Bolzano-Weierstrass Theorem, stating that every bounded sequence has a convergent subsequence.

Derivable Functions: Limits and Continuity

Explores derivable functions, limits, continuity, and differentiability, emphasizing the relationship between them.

Subsequences: bounded sequences and Bolzano-Weierstrass theorem

Introduces bounded sequences and subsequences, culminating in the Bolzano-Weierstrass theorem.

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.