Calculus of Variations: Principles and Applications

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Page 3 of 4

Subsequences and Bolzano-Weierstrass Theorem

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

Advanced Analysis I: Continuous Functions on Compact Sets

Explores the necessity of uniform continuity for continuous functions on compact sets.

Sequence Convergence: Definitions and Properties

Covers definitions and properties of sequence convergence, including limits, uniqueness, and the two gendarmes theorem.

Convergence Criteria: Polynomials Limits

Covers convergence criteria for polynomial limits and the theorem of the two gendarmes, emphasizing the importance of indeterminate forms.

Limit of a Sequence

Explores the limit of a sequence and its convergence properties, including boundedness and monotonicity.

Limits of Sequences

Explores the concept of limits of sequences and their convergence towards infinity or negative infinity.

Generalized Integrals: Bounded Interval

Introduces generalized integrals on a bounded interval, discussing convergence, divergence, comparison criteria, variable substitution, and corollaries.

Stone-Weierstrass Theorem

Explores the Stone-Weierstrass theorem, proving uniform density of specific function families on compact sets.

Multiple Integration: Part 2

Explores double integrals over unbounded domains and convergence of sequences.

Analysis IV: Convergence and Approximation in L² Space

Explores convergence and approximation in L² space, emphasizing the limitations of continuous functions and the importance of closed sets.