Analysis
Checkpoints
- Open and Closed Balls Basis Introduces open and closed balls in a metric space — the precise, distance-based zones that replace the intuitive notion of 'nearby points'.
- e (Base of Natural Logarithm) Basis Introduces Euler's number e ≈ 2.71828, defined as the limit of compound growth, shown to equal an infinite series of reciprocal factorials, and proved irrational.
- Limits of Sequences Basis Defines the limit of a sequence in a metric space, proves that limits are unique, and explores the equivalent neighborhood characterization.
- Neighborhood Basis Defines neighborhoods of a point in a metric space and shows how they generalize open balls into a flexible, radius-free language for closeness.
- Real Number (By Rational Number Closure) Basis Defines the real numbers by closing up the gaps in the rationals: Cauchy sequences of rationals are grouped into equivalence classes to form R, the unique complete ordered field in which Q is dense.
- Supremum & Infimum Basis Defines the supremum (least upper bound) and infimum (greatest lower bound) of a subset of ℝ, establishes the least upper bound property as the completeness of ℝ, and derives key consequences including the Archimedean property.