∫

∑

∂

∞

√

∏

Home/Calculus/Chapter 2

Back to Calculus

Chapter 2

Available

Sequence Limits

Master the rigorous ε-N definition of limits, fundamental convergence theorems, and the theoretical foundations connecting to real number completeness

4 Courses

14-18 Hours

Intermediate Level

What You'll Learn

Rigorous ε-N definition of sequence limits

Fundamental convergence theorems with complete proofs

Connection to real number completeness

Interactive practice with step-by-step solutions

Prerequisites

Set Theory Foundations (CALC-1.1)

Real Number Construction (CALC-1.2)

Properties of Real Numbers (CALC-1.3)

Functions Fundamentals (CALC-1.4)

Complete Chapter 1 first →

Courses in This Chapter

CALC-2.1

Sequences and the ε-N Definition

Master the precise ε-N definition of sequence limits, equivalent formulations, and divergence criteria

3-4 hours

Sequence Definition

ε-N Definition

Neighborhood Formulation

Divergent Sequences

CALC-2.2

Properties and Operations of Limits

Prove uniqueness of limits, boundedness, sign preservation, and master arithmetic operations on limits

Fundamental Convergence Theorems

Apply the Squeeze Theorem, Monotone Bounded Theorem, derive the limit e, and use Stolz theorem

4-5 hours

Squeeze Theorem

Monotone Bounded Theorem

The Number e

Stolz Theorem

CALC-2.4

Subsequences, Completeness, and Cauchy Criterion

Understand subsequences, Bolzano-Weierstrass theorem, Cauchy criterion, and completeness equivalences

Historical Context

The rigorous definition of limits emerged in the 19th century as mathematicians sought to place calculus on firm logical foundations. This chapter covers the foundational work of:

Augustin-Louis Cauchy

Rigorous limit definitions

Karl Weierstrass

ε-δ formalization

Bernard Bolzano

Bolzano-Weierstrass theorem