Revised 1/2026

MTH 295 - Introduction to Proofs and Real Analysis (3 CR.)

Course Description

The purpose of this course is to help students transition to more advanced topics in Mathematics.  The topics covered in the course will include Propositional Logic, Set Theory, Relations and Functions, Cardinality, and Introductory Real Analysis (Advanced Calculus).  This course will emphasize proof writing, and students will learn the mathematical typing setting language LaTeX which they will use to write their homework for class.

General Course Purpose

This course is for students who plan on taking more advanced Mathematics courses at a 4-year college or university.  This course will ensure students are ready to take more advanced courses in Mathematics immediately after they transfer to their 4-year school.

Course Prerequisites/Corequisites

Prerequisite: MTH 264 or equivalent.

Course Objectives

• Proof writing techniques, including Direct Proof, Proof by Contradiction, Proof by Contrapositive
• Axiomatic Set Theory and Russel’s Paradox
• Propositional Logic
• Equivalence of Well Ordering, Weak Induction, and Strong Induction on the Natural Numbers.
• Relations, including equivalence relations and partitions, and ordering relations and partial orderings
• Functions, including Injective, Surjective and Bijective Functions
• Cardinality of Sets, including Finite Sets, Infinite Sets, Countable Sets, Uncountable sets
• Cantor-Schroeder-Berstein, Cantor’s Theorem and the Axiom of Choice
• Cauchy’s Construction of the Real Numbers and the Completeness of the Real Numbers
• Heine-Borel Theorem
• Bolzano-Weierstrass Theorem
• Equivalent forms of Completeness
• Limits and Continuity, including The Intermediate Value Theorem