Mathematics and Statistics

Syllabus for Analysis

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