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## Syllabus for Analysis

Link to individual course syllabus:

The real numbers:

Mathematical induction

Rational and irrational numbers

Real numbers and the completeness axiom

Sequences:

The convergence of bounded, monotone sequences of reals

Cauchy sequences

Bolzano-Weierstrass Theorem (for bounded sequences)

Point-set theory:

Countable and uncountable sets

Accumulation points (also called cluster points or limit points)

Bolzano-Weierstrass Theorem (for bounded infinite sets)

Open and closed sets

Compact sets

Heine-Borel Theorem

Continuous functions:

Intermediate Value Theorem

Basic properties of functions continuous on a compact set:

Boundedness

Attainment of extreme values

Uniform continuity

Continuity of sums, products, quotients of continuous functions

Differentiability and derivatives:

Limit definition of derivative

Derivatives at local extreme points

Rolle's Theorem, Mean Value Theorem

Integration:

Definition of Riemann integral

Integrability of a continuous function over *[a, b]*

Infinite series:*p*-series and geometric series

Absolute and conditional convergence

Comparison, ratio, and alternating series tests

Uniform convergence:

Continuity of the limit function

Integration of sequences and series

Differentiation of sequences and series

Weierstrass *M*-test

Proving uniform convergence (or lack thereof) in specific examples

Power series:

Radius and interval of convergence, behavior at endpoints

Continuity, differentiation, and integration of power series

Definitions and statements of the basic theorems are important