Ian Charlesworth

Math 104 - Introduction to Analysis

Midterm post-mortem

Generally, the distribution of midterm grades was around what I expected it to be and around what I was aiming for. I was surprised by the difficulty of problem 5. Problems 2 and 3 seemed to be the least useful: they had the lowest variance of scores assigned, and the lowest covariance with total score. Problem 5 had the highest covariance with final score, closely followed by Problem 6.

Here are some comments and observations about the midterm and common errors:

• You can't quantify over an expression. You can't say "for every $\frac\epsilon2 \gt 0$", for the same reason you can't say "for every $\sin(x) \gt 0$". You can say "for every $\epsilon \gt 0$" and then use $\frac\epsilon2$, or "for every $x$ for which $\sin(x) \gt 0$".
• Labels are inconsequential. The definition of Cauchy is the same no matter what letters you use. If you let $\alpha \gt 0$ and eventually show that there is some $Q$ so that whenever $s, t \gt Q$ you have $d(a_s, a_t) \lt \alpha$, you don't need to write "now set $\epsilon = \alpha$ and $N = Q$ and we see that $(a_n)_n$ is Cauchy"; you're already done. Likewise, there's no need for this bound to be a "single letter": if you show that the distance estimate holds once $s, t \gt \sqrt{55+\alpha^{-5} + 4\alpha}$, you're done; you don't need to introduce some $N = \sqrt{55+\alpha^{-5}+4\alpha}$.
• Relatedly, symbols only have meaning if you give it to them. If you say something is continuous, that doesn't automatically imply anything about some symbols $\epsilon$ and $\delta$ unless you do it explicitly; just because they appear in the definition or in a statement of a conditional does not carry that meaning beyond. As something of an example of what I mean, consider the following incorrect argument. "For any $x \in \R$ there is some even $N \in \Z$ with $N \gt x$. Also, for any $x \in \R$ there is some odd $K \in \Z$ with $K \lt x$. Now $K \lt x \lt N$, so..." Even worse, consider the following. "For any $x \in \R$ there is some even $N \in \Z$ with $N \gt x$. Also, for any $x \in \R$ there is some odd $N \in \Z$ with $N \lt x$. But now $N$ is both odd and even, a contradiction!"
• For whatever reason it is conventional to use the letter $F$ to refer to a closed set, and either $G$ or $U$ for an open set. This is similar to how $n$ is usully an integer; you could let $n \in \R$, but it would be weird. You can let $F \subset M$ be open, but it looks a little weird.
• Negating logical statements can be tricky, and is probably worth practicing. Several people messed up negations of Cauchy or of compact. Remember that if "for all $x$ something happens" is false, that means "for at least one $x$ it doesn't", not "for all $x$ it doesn't happen".
• I commented a few times but wound up not taking off points for the mistake of assuming that an open cover must be countable. That is, an arbitrary open cover can be written $\{G_i \mid i \in \mathcal{I}\}$ for some set $\mathcal{I}$, but you can't necessarily write it as $\{G_1, G_2, \ldots\}$ since it may have too many sets in it to be written that way. It's generally better to not name the sets in the open cover; you can just write "let $\mathscr U$ be an open cover" instead of "let $\mathscr U = \{G_i \mid i \in \mathcal I\}$ be an open cover". This is also nice because in the first case you can write an arbitrary finite subset as $\{G_1, \ldots, G_n\} \subseteq \mathscr U$, whereas in the second you need to "choose $i_1, \ldots, i_n \in \mathcal I$ and take $\{G_{i_1}, \ldots, G_{i_n}\} \subseteq \mathscr U$" which is more complicated.
• The word "Let" is a little tricky because it plays two subtly different roles: either allowing some label to be an arbitrary thing, or defining some label as meaning something specific. For example, both "Let $x$ be an integer" and "Let $x = \frac34$" are valid things to say. It is not valid to write "Let $x = \frac34$ be an integer." It's not even valid to write "Let $x = 3$ be an integer."
• Many people got confused about the definition of cluster points. If $S \subseteq M$, any point of $M$ could potentially be a cluster point of $S$, not only points of $S$.
• Don't feel the need to introduce variables for the sake of introducing variables; it often makes the argument more cluttered (making it harder to keep striaght yourself, and harder to read) without adding anything. For example, don't write something like "Let $\epsilon = \frac12$." There's already a symbol to denote $\frac12$, it's $\frac12$.

Here is some data about how grades are shaping up so far.