Ian Charlesworth

Math 104 - Introduction to Analysis

Midterm post-mortem

There were not too many common mistakes on the midterm, and I tried to give a lot of individual feedback when grading it so you should take a look on Gradescope. Below I've noted a few common problems, and some stylistic notes too.

• One common mistake was to apply the triangle inequality within an absolute value, writing something like \[\abs{d(x, y) - 3} \leq \abs{d(x, z) + d(z, y) - 3}.\] This can be false if, for example, $d(x, y) = d(x, z) = d(z, y) = 1$. Mostly this happened on problem 2, and the inquality that was claimed was incidentally true, but not supported by the argument made.
• Relatedly, there were at least a few people who at some point or another went from $x \lt y$ to $\abs x \lt \abs y$ or from $\abs x \lt \abs y$ to $x \lt y$.
• Many people got confused about the definition of limit points. If $S \subseteq M$, any point of $M$ could potentially be a limit point of $S$, and points in $S$ are not necessarily limit points.
• Relatedly, points in $\overline{S}$ are *also* not necessarily limit points. Points in $\overline{S}$ with sequences in $S$ converging to them are not necessarily limit points.
• 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 $\frac12$.
• Likewise, it's probably simpler to write $\gt$ rather than $\not\leq$. There are relations, like $\subseteq$, where $\not\subseteq$ is not the same as $\supset$, but since orderings are total this isn't the case.
• There were a few people who did some weird things with quantifiers which ranged from incorrect to merely very strange wording. For example, writing things like "$\exists x = 5 \in \R$". This is not good, because it is trying simultaneously to specify $x$ and to let it be arbitrary. You should write either "Let $x = 5$" or "$\exists x \in \R$" (or I suppose if you really want you could write something like "$\exists x \in \R$ such that $x = 5$").
• 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."
• There were a few cases of people reusing variables in strange ways, or trying to requantify things that were already quantified. You can't, for example, say something like "Let $x = 4$. Let $f : \R \to \R$ be defined by $f(x) = x^2$."

Many people did well on the exam, and there was a cluster near the top. On the final, I plan to make the easiest question easier and the hardest question harder.

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