Math 3110
Syllabus
is
here.
Textbook
Understanding Analysis by Abbott, available as a PDF through the Cornell library.
Resources
Here is a link to a textbook that has been used for Math 3040 in the past.
This appendix to an old 3110 textbook might be a useful resource for those new to proofs.
The axioms for a complete ordered field.
Homework
You do not have to type your homework with LaTeX, though it's a nice skill to learn. If you choose to (neatly!) handwrite your homework, you should scan it to Gradescope using e.g. the builtin document scanner that comes with Box or Dropbox. There are other good smartphone scanning apps, or you can use a traditional scanner at the library.
Guidelines for submitting homework:

You should submit a PDF, not an image file, to Gradescope.

You must identify for Gradescope which portion of your file corresponds to which problem. Your homework will not be accepted until this is done.

When you start a new problem, consider starting it on a new page.

Genius Scan is a program some of your classmates have recommended. The documentscanner feature on Dropbox also works well.
Homework files:

Homework 12, due Wednesday, 8 May, on Gradescope. (LaTeX source)

Homework 11, due Wednesday, 1 May, on Gradescope. (LaTeX source)

Homework 10, due Wednesday, 24 April, on Gradescope. (LaTeX source)

Homework 9, due Monday, 15 April, on Gradescope. (LaTeX source)

Homework 8, due Friday, 29 March, on Gradescope. (LaTeX source)

Homework 7, due Wednesday, 20 March, on Gradescope. (LaTeX source)

Homework 6, due Wednesday, 13 March, on Gradescope. (LaTeX source)

Practice problems for Prelim 1 (not to be turned in)

Homework 5, due Friday, 1 March, on Gradescope. (LaTeX source)

Homework 4, due Wednesday, 20 February, on Gradescope. (LaTeX source)

Homework 3, due Wednesday, 13 February, on Gradescope. (LaTeX source)

Homework 2, due Wednesday, 6 February, on Gradescope. (LaTeX source)

Homework 1, due Wednesday, 30 January, on Gradescope. (LaTeX source)
Schedule
 Mon 6 May: Review of the course. What comes next?
 Fri 3 May: Proofs of the addition formulas for sine and cosine.
Proof of the existence of \(\pi\).
 Wed 1 May: Termbyterm differentiation of power series (roughly Theorems 6.5.6–6.5.7 in the textbook.) The exponential function has a powerseries expansion.
 Mon 29 Apr: Power series. Radius of convergence and absolute convergence on the interior of the interval of convergence. The Root Test. (§6.4–§6.5, roughly.)
 Fri 26 Apr: Uniform limits of integrable functions are integrable. Series of functions, the Weierstrass MTest. (§6.4 and Theorem 7.4.4.)
 Wed 24 Apr: Uniform limits of continuous functions are continuous.
Uniform limit of derivatives must be the derivative of the limit function. (§§6.2–6.3.)
 Mon 22 Apr: Sequences and series of functions. Pointwise vs. uniform convergence. (§6.2.)

Fri 19 Apr: The Fundamental Theorem of Calculus. (§7.5.)

Wed 17 Apr: Example (\(\int_0^1 x^2\,dx = 1/3\)). Basic properties of the integral. (mostly Theorem 7.4.2.)

Mon 15 Apr Apr: Review for prelim by proving some basic properties of Lipschitz functions.

Fri 12 Apr: More on integrability, basic properties of the integral. (§§7.3–7.4.)

Wed 10 Apr: Integrability. Integrability criterion. Continuous functions are integrable. (§7.2)

Mon 8 Apr: Introduction to integration. Partitions. (§§7.1–7.2)

Fri 29 Mar: Finish derivatives. Taylor's Theorem.

Wed 27 Mar: Fermat's Theorem (derivative zero at local min/max). Mean Value Theorem. Derivatives are "Darboux," if time allows. (§§5.2–5.3)

Mon 25 Mar: Constructor theorems for differentiable functions and their proofs: Product Rule, Quotient Rule, Chain Rule. (§5.2)

Fri 22 Mar: Continuous functions preserve functional limits. Derivatives: definitions, examples, differentiability implies continuity. (§5.2)

Wed 20 Mar: Continuity on a compact set implies uniform continuity. Functional limits: definition, examples, constructor theorem. (§§4.4 and 4.2)

Mon 18 Mar: Uniform continuity: definition and examples. Started uniform continuity on compact sets. (§4.4)

Fri 15 Mar: Proof of the Boundedness Theorem for functions on a closed interval. (Different from the book's proof.) Sequential characterization of continuity. Second proof of Boundedness Theorem for compact sets using sequential continuity. (§§4.3–4.4.)

Wed 13 Mar: Proof of the Intermediate Value Theorem. (Note: we gave a different proof than the book gives.) Started proof of the Boundedness Theorem.

Mon 11 Mar: Comments about and examples/nonexamples related to the definition of continuity. Constant and linear functions are continuous. Sums and products and quotients of continuous functions are continuous; we proved only the fact for sums. (§4.3)

Fri 8 Mar: Introduction to continuous functions with three examples. (§4.1.)

Wed 6 Mar: Proof of Heine–Borel Theorem, partial proof that \(\sqrt{2}\) exists. (§3.3, but we didn't follow the book's treatment)

Mon 4 Mar: Finished characterization of compactness. (§3.3.)

Fri 1 Mar: Closure and complements, compactness. (§3.2–§3.3.)

Wed 27 Feb: Open and closed sets, cluster points. (§3.2.)

Fri 22 Feb: Introduction to the basic topology of \(\mathbf{R}\): the Cantor MiddleThirds Set, open and closed sets. (§§3.1–3.2.)

Wed 20 Feb: Final comments about series: Comparison Test, Alternating Harmonic Series, absolute and conditional convergence, maybe some comments about rearrangements. (§2.7)

Mon 18 Feb: Finished Cauchy criterion; geometric series, other comments about series. (§§2.6–2.7)

Fri 15 Feb: Introduction to series. Cauchy sequences if time allows. (§§2.4, 2.7)

Wed 13 Feb: subsequences, examples, the Bolzano–Weierstrass Theorem. (§2.5)

Mon 11 Feb: finished the uncountability of the reals. Monotone Convergence Theorem for sequences and an example. (§1.5, §2.4)

Fri 8 Feb: cardinality, the countability of the rationals and the uncountability of the reals (§1.5)

Wed 6 Feb: loose ends: uniqueness of limits, sequential density of rationals in reals, more on algebraic properties of limits (§2.2–2.3)

Mon 4 Feb: algebraic and order properties of limits, boundedness of convergent sequences [substitute] (§2.3)

Fri 1 Feb: sequences and convergence of sequences, examples [substitute] (§2.2)

Wed 30 Jan: the Archimedean Principle, the orderdensity of the rationals (mostly §1.4; see the Appendix linked above too)

Mon 28 Jan: Axiom of Completeness (§1.3)

Fri 25 Jan: background: sets, functions, and induction (§§1.1–1.2)

Wed 23 Jan: Intro, welcome, §§1.1–1.2