Homework

Homework is due every Wednesday at 10:00 AM on [Gradescope], one hour before class starts. (Use this invite code if you’re not signed up). Please read the late homework policy

Mostly correct solutions that were turned in by students will be posted when the assignment is graded. Disclaimer: These solutions are not always perfect, but are good enough to show you how the problem can be done. You should understand these solutions thoroughly. If similar questions appear on exams, you shouldn’t blindly reproduce these solutions; but write a completely correct solution after filling in gaps and fixing minor errors.

Homework 1 (due 2026-01-21)

1. Find the discussion board, and post your favorite math fact. (Include a screenshot when you turn in your homework.)
2. Prove this proposition ($L^p$ distance is a metric).
3. Prove the second part of this example (closed set examples).
4. 1. Prove the second part of this proposition (union of open sets).
   2. Do the first part of this problem (intersection of open sets).
5. Do this problem ($n^\text{th}$-root continuous).
6. 1. Prove this proposition (limit of products).
   2. Prove this proposition (limit of quotients).

Mostly perfect student solutions

Homework 2 (due 2026-01-28)

1. 1. Prove this proposition (boundary of a set).
   2. Do this problem (limit not existing example).
2. 1. Do problem this problem (extended reals).
   2. Do problem this problem (limits at infinity).
3. 1. Do this problem (limit along lines).
   2. Do this problem (deciding continuity using curves).
4. Do this problem (rational functions).
5. Do this problem (4 examples).

Mostly perfect student solutions

Homework 3 (due 2026-02-04)

1. 1. Prove this theorem (inverse image of open sets).
   2. Do both parts of this problem (forward image of open sets).
2. 1. Prove the existence of integer part of reals.
   2. Prove the density of rationals.
   3. Prove the density of irrationals.
   4. Prove the existence of decimal expansions.
3. Prove the intermediate value theorem.
4. Prove $\R^d$ is connected.
5. Show that any odd polynomial has a root.

Mostly perfect student solutions

Homework 4 (not due)

In light of your Midterm this homework is not due. The problems address material that will be on the syllabus, so I recommend thinking about them. No solutions will be posted, other than the hints given in class/recitations, and in the problems.

1. Prove the Riemann series theorem
2. Prove the ratio test
3. Prove the Bolzano Weierstrass theorem for $d = 1$, without using the rising sun lemma.
4. Prove continuity of inverses
5. Prove uniform continuity on compact sets

Homework 5 (due 2026-02-18)

1. 1. Proposition 29 (limit of $f’$)
   2. Intermediate value theorem for derivatives
2. 1. This proposition characterizing differentiable convex functions.
   2. This problem showing convex functions lie above their tangent
   3. This problem describing convex functions.
3. 1. Taylor’s theorem (first version)
   2. Taylor’s theorem (second version)
4. Prove the derivative tests.

Homework 6 (due 2026-02-25)

1. This problem showing partials existing doesn’t imply differentiability.
2. 1. This proposition differentiating vector functions.
   2. This proposition showing a product rule.
3. This problem computing the (one variable) derivative.
4. Polar coordinates (all 3 parts). (For practice do Examples 30, 31, and Problem 32 but don’t turn them in.)
5. This problem showing mixed partials aren’t always equal.
6. This problem about divergence and curl.