Proof — Intermediate

Quantifiers, ε-δ arguments, equivalences, and sets

About this section

The Intermediate section of Proof studies the proof techniques required at the university level: a deep treatment of quantifiers, the ε-δ formulation of limits and continuity, proofs of equivalent propositions, multi-case analysis, and arguments about sets and mappings.

These techniques form the language in which analysis, algebra, and other higher mathematics are written.

Contents

Chapter 1: Proofs Involving Quantifiers

How to prove propositions involving the universal symbol ∀ and the existential symbol ∃.

Chapter 2: ε-δ Arguments (Part 1) — Definitions and Basic Examples

Rigorous definitions and basic examples for quantifiers, limits, continuity, and sequence limits.

Chapter 2: ε-δ Arguments (Part 2) — Pitfalls and Extensions

Common mistakes, proofs of non-existence of limits, comparison of four proof templates, uniform continuity, and exercises.

Chapter 3: Proofs of Equivalence

Techniques for proving "necessary and sufficient conditions" and "A and B are equivalent."

Chapter 4: Case Analysis

Techniques for proofs that split into multiple cases, and how to verify exhaustiveness.

Chapter 5: Sets and Logic

The relationship between set operations and logical operations, and proof techniques about sets.

Chapter 6: Exercises

A capstone of the Intermediate section with a variety of proof problems.

Learning goals

  • Accurate understanding and proofs of propositions containing quantifiers.
  • Rigorous proofs of limits via the ε-δ method.
  • Mastery of the structure of equivalence proofs (necessary and sufficient conditions).
  • Effective design and execution of case analysis.
  • Techniques for proofs about sets (inclusion, equality).

Frequently Asked Questions

What does the Intermediate section of Proof cover?

It covers the handling of quantifiers (∀, ∃), rigorous proofs of limits via the ε-δ method, proofs of equivalent propositions, multi-case analysis, and proofs about sets and mappings. These form the foundation of university-level analysis and algebra.

What is the difference between Basic and Intermediate proofs?

At the Basic level, the focus is on deriving a specific conclusion from specific assumptions via direct computation. At the Intermediate level, one adds techniques for propositions with multiple quantifiers (such as "for every ε there exists a δ") and for more abstract structures such as set equalities and properties of mappings.

How are intermediate proof techniques used in real mathematics?

ε-δ arguments are the foundation of every rigorous proof in analysis (continuity, differentiation, integration, convergence). Proof techniques for sets and mappings appear in algebra, point-set topology, and functional analysis. Reasoning about quantifiers also connects directly to logic and specification proofs in computer science.