1. History of analysis

1.2. The Age of Rigor and the Real Numbers (1800s)

1. The Demand for Precision

  • Cauchy introduced limits, continuity, and convergent sequences using informal definitions.

  • Weierstrass formalized these ideas using precise ε-δ (epsilon-delta) arguments, eliminating reliance on intuition.

2. Constructing the Real Line

  • Mathematicians realized that a rigorous understanding of the real numbers was essential.

  • Dedekind introduced Dedekind cuts to construct real numbers from rationals.

  • Cantor used Cauchy sequences to complete ℚ into ℝ, formalizing the notion of completeness.

3. Early Integration Theory

  • Riemann defined integration via finite sums over partitions, forming the Riemann integral.

  • This worked well for many functions but had limitations—especially for discontinuous or unbounded functions.