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Basic and Real Analysis
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Basic and Real Analysis
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About course
Study of proofs of mathematical principles
Syllabus
Section 1. Sets and Metric Spaces
The Least Upper Bound Property
Constructing the Rational Numbers
Properties of Q
Construction of the Reals
Limit Points
Cantor Diagonalization and Metric Spaces
Countable and Uncountable Sets
Principle of Induction
Complex Numbers
The Relationship Between Open and Closed Sets
Compact Sets
Relationship of Compact Sets to Closed Sets
Compactness and the Heine-Borel Theorem
Connected Sets, Cantor Sets
Section 2. Sequences and Series
Convergence of Sequences
Subsequences, Cauchy Sequences
Complete Spaces
Series
Series Convergence Tests, Absolute Convergence
Section 3. Functions and Continuity
Functions - Limits and Continuity
Continuous Functions
Uniform Continuity
Discontinuous Functions
Section 4. Derivatives
The Derivative and the Mean Value Theorem
Section 5. Taylor's Theorem
Taylor's Theorem, Sequence of Functions
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