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Assignment 2

AMATH 331/PMATH 331

Due: 11:59 p.m. October 18th, 2023

This assignment roughly covers topics from lectures 7 - 12, covering from Cauchy Sequences to Properties of Closed Subsets of Rn.

You may (but do not have to) discuss ideas and strategies for solutions with your classmates, and I encourage this collaboration. You can also use office hours to get help from the instructor or the TAs. However, your submitted solution should be written up yourself, in your own words: this means NO DIRECT COPY-PASTING solutions from your classmates. This also means that you do not want to copy-paste everything from the collaborations into your scrap work/draft: you may jot down key ideas from the collaboration and understand (in your own words) how those ideas connect in various levels of details, let it sink for a few hours at least, then write up your solutions when you are alone by filling up the details.

You need to justify all your work by default, unless you are asked not to. For example, if a questions asks you to provide an example (or counterexample), then you also need to show why your example is indeed a requested example. Also, the general rule of thumb is cite the theorems/lemmas with special names (e.g., LUBP, MCT, etc), but there are few special named theorems that don’t need citations (e.g., Arithmetic Rule for Limits). If you are unsure, please ask me (the instructor).

You may prepare your answers digitally or handwritten on paper. If prepared on paper, you can scan or take photos of each page for submission. Please ensure that the images, and your handwriting, are clearly legible and in the correct orientation. You can use apps to process and enhance photos for clarity. Please also separate submissions by problem number in Crowdmark.

Problem 1

Consider a sequence (an)∞n=1 of real numbers.

(a) Suppose that |an+1 − an| ≤ n 12 for every n. Show that (an)∞n=1 converges.

Hint: In this case, is (an)∞n=1 Cauchy?

(b) Suppose instead that (an)∞n=1 is Cauchy. Show that there is a subsequence (ank )∞k=1 such that

|ank+1 − ank | ≤1k 2 for each k.

Problem 2 (More Examples of Complete Subsets of R)

(a) Prove, by directly using the definition of the limit of a sequence, that if L is a real number and (an)∞n=1 is a convergent sequence of real numbers such that an ≤ L for every n, then

limn→∞an ≤ L.

Hint: Try to prove it by contradiction.

(b) Using part (a) above and the definition of complete subsets of R, show that for every L ∈ R, the half-closed interval (−∞, L] is complete. For this problem, DO NOT USE the Proposition that a subset of R n is closed iff it is complete.

Problem 3

Either prove it or provide a counterexample to each of the following statements:

(a) If P ∞n=1 an converges, then P ∞n=11an converges.

(b) If P ∞n=1 an converges and (bn)∞n=1 converges, then P ∞n=1 anbn converges.

(c) If P ∞n=1 an converges conditionally, then P ∞n=1 n 2an diverges.

Problem 4

(a) Prove the parallelogram law: for every ⃗x, ⃗y ∈ R n,

∥⃗x + ⃗y∥ 2 + ∥⃗x − ⃗y∥ 2 = 2∥⃗x∥ 2 + 2∥⃗y∥ 2 .

(b) Using the definition for an angle θ between two vectors ⃗x, ⃗y ∈ R n, prove the cosine law:

∥⃗x + ⃗y∥ 2 = ∥⃗x∥ 2 + 2∥⃗x∥∥⃗y∥ cos θ + ∥⃗y∥ 2 .

Problem 5

(a) Prove that if (⃗xk)∞k=1 is a sequence of points in R n that converges to ⃗a ∈ R n, then

limk→∞∥⃗xk∥ = ∥⃗a∥.

(b) Prove that if a sequence (⃗xk)∞k=1 of points in R n satisfies

∞Xk=1∥⃗xk − ⃗xk+1∥ < ∞,

then (⃗xk)∞k=1 is Cauchy.

Problem 6

The goal of this problem is to prove more examples of closed subsets of R n for n ≥ 2.

(a) Show that any unit vector ⃗u ∈ R n (i.e., ∥⃗u∥ = 1) is a limit point of the open unit ball B1(0) := {⃗x ∈ R n : ∥⃗x∥ < 1}.

Hint: Draw a line passing through ⃗u and the origin 0, and choose a sequence of points in the intersection of this line and B1(0).

(b) Using Problem 2(a), 5(a), and 6(a) from above, show that the unit disk D := {⃗x ∈ R n : ∥⃗x∥ ≤ 1} is equal to the closure B1(0) of the open unit ball.

Remark: This implies that the unit disk D is closed, because the closure of any set is closed.

(c) Prove that the set X := {(x, y) ∈ R 2 : xy = 1} is closed.