Homework Assignment #6

Warning. An unusually difficult assignment; constructive cooperation is highly encouraged! (Though as always, you'll lose if you don't struggle some on your own).

Read sections 27 and 37 in Munkres' textbook (Topology, 2nd edition), and if curious, also sections 9-11. Remember that reading math isn't like reading a novel! If you read a novel and miss a few details most likely you'll still understand the novel. But if you miss a few details in a math text, often you'll miss everything that follows. So reading math takes reading and rereading and rerereading and a lot of thought about what you've read. Also, preread sections 30 through 33, just to get a feel for the future.

Solve and submit problem 2 on page 177 of Munkres book, as well as the following

Additional Problem. Let $S$ be the set of all bounded sequences of real numbers, and for each $a\in S$ let $M_a\geq 0$ be its least upper bound: the least number for which $\forall k\,|a_k|\leq M_a$. Let $X:=\prod_{a\in S}[-M_a,M_a]$, let $\alpha\colon\bbN\to X$ be defined by $\alpha(n)_a=a_n$, and let $\beta\bbN$ be the closure of the image of $\bbN$ in $X$ via $\alpha$: meaning, $\beta\bbN:=\overline{\alpha(\bbN)}$.

1. Prove that $\beta\bbN$ is a compact space and that $\alpha(\bbN)$ is a dense homeomorphic copy of $\bbN$ inside $\beta\bbN$ (here $\bbN$ is taken with the discrete topology). Conclude that $\beta\bbN\neq\alpha(\bbN)$.
2. Prove that for any bounded sequence $a\in S$, there exists a unique continuous function $\bar{a}\colon\beta\bbN\to[-M_a,M_a]$ such that $a=\bar{a}\circ\alpha$.
3. Fix some $\mu\in\beta\bbN\setminus\alpha(\bbN)$ and define $\Lim_\mu(a):=\bar{a}(\mu)$. Prove that $\Lim_\mu$ is the bestest ever definition of a "limit of a sequence": it is always defined, if a sequence $a$ has a limit in the ordinary sense then $\lim a_k=\Lim_\mu(a)$, and $\Lim_\mu$ is linear and multiplicative: for any pair of bounded sequences $a$ and $b$ and any scalars $A,B\in\bbR$, $\Lim_\mu(Aa+Bb)=A\Lim_\mu(a)+B\lim_\mu(b)$ and $\Lim_\mu(ab)=\Lim_\mu(a)\Lim_\mu(b)$.
4. Yet $\Lim_\mu$ fails in (at least) one way: show that if $a$ is a bounded sequence and $b$ is obtained from $a$ by "a shift to the left" (namely, $b_k=a_{k+1}$), then it isn't always the case that $\Lim_\mu(a)=\Lim_\mu(b)$.
5. Let $b$ (for blink) be the sequence $(0,1,0,1,0,1,\ldots)$. What can you say about $\Lim_\mu(b)$?

In addition, ponder the following question (but do not write or submit anything): Where oh where in MAT157/257, was "uniform continuity" extensively used, and why was mere continuity insufficient there.

In addition, if you're in the mood, do all the "supplementary exercises" on Munkres pages 72-74 (but don't submit anything, of course).

Due date. This assignment is due at the end of class on Thursday, November 15, 2018. New! If you can, please use the Homework Submission Cover Page to help with faster returns and to help with privacy.

Dire Warning. Right after Reading Week agents of the Evil Galactic Empire will lock all the students of this class in separate sound proof, electromagnetically sealed, neutrino hardened, and gravitational wave resistant rooms in the dark, cold lower basement of Sidney Smith Hall. In the rooms they will place identical countable sequences of numbered boxes, each one containing a real number (the same sequence of real numbers in each room). By Tuesday morning, each student must open all but one of their boxes in the order of their liking, and guess the number in the remaining box. If more than one student will guess wrong [oh no, redacted].
Do Something! You must devise a survival strategy over reading period or else we will never study the absolutely stunning Urysohn's Lemma!

("Saw Omega" from Alfonso Gracia-Saz from Mira Bernstein from Vigorous Handwaving [spoilers inside]. Deadly serious.)