Introduction to Real Analysis

Rutgers Math 311H, Fall 2025
Professor: Alex Kontorovich

Course Description: The primary aim of this course is to teach you the rigorous foundations of calculus — foundations which were omitted in earlier calculus courses. We will give precise definitions of the concept of a limit and of such concepts as the continuity and differentiability of functions, and will prove some of the many consequences which may be deduced from these definitions. A secondary aim is to improve your skills in reading and understanding mathematics written by others and in writing precise definitions and rigorous proofs of your own.

More information: is available on the Course Info Page.

Course Resources

Week Lecture 1 Video Lecture 2 Video Topics/Content/Notes/Other
Week 1, 09/02, 09/05 Lecture 1 Notes
Newton, Leibniz, Cauchy, Fourier,
Lean tactics: exact, rfl, rewrite,
ring_nf, use, intro,
specialize, choose

Lecture 2 Notes
Newton's computation of π
Formal definition of the limit of a sequence
Week 2, 09/09, 09/12 Installing Lean locally (VS Code etc), GitHub (Issues, Pull Requests) Lecture 3 Notes
Archimedean Property, Casting/Coercion,
bound, linarith,
field_simp, exact_mod_cast

Week 3, 09/16, 09/19 MIDTERM 1 REVIEW, and
Lecture 4 Notes
Non-convergence of (-1)^n, triangle inequality.

Lecture 5 Notes
Doubling a sequence doubles the limit.
Week 4, 09/23, 09/26 MIDTERM 1 Lecture 6 Notes
Limit of a Sum is the the Sum of the Limits,
dealing with conjunction/disjunction (And and Or) in the Goal and Hypotheses, and
the Squeeze Theorem
Week 5, 09/30, 10/03 Lecture 7 Notes
Limit is Unique, Convergent Sequence stays away from zero,
Limit of Abs Value is the the Abs Value of the Limit,
Limit of Reciprocal is Reciprocal of the Limit

Lecture 8 Notes
The `by_cases` Tactic, Induction
Week 6, 10/07, 10/10 Lecture 9 Notes
Construction of Naturals (Induction), Finite Sums, Convergent implies Bounded

Lecture 10 Notes
Limit of products is product of Limits; the Algebraic Limit Theorem
the Order Limit Theorem; Subsequences
If Convergent, any Subsequence has same limit
Week 7, 10/14, 10/17 Lecture 11 Notes
Cauchy Sequences!
If a sequence has a limit, then it's Cauchy;
The sum of Cauchy sequences is Cauchy;
If a sequence is Cauchy, then it's bounded.

Lecture 12 Notes
If a sequence is Monotonic (non-decreasing) and Bounded, then it is Cauchy!
Week 8, 10/21, 10/24 Lecture 13 Notes
Orbits, "Fancy" Choose, Peaks, Unbounded Peaks, and Monotone Subsequences.

Week 9, 10/28, 10/31 MIDTERM II
Week 10, 11/04, 11/07
Week 11, 11/11, 11/14
Week 12, 11/18, 11/21
Week 13, WED 11/26 (FRI schedule) HAPPY THANKSGIVING!
Week 14, 12/02, 12/05
Week 15, 12/09 FINAL REVIEW