Ian Charlesworth

Math 104 - Introduction to Analysis

Midterm 2 post-mortem

This was a challenging midterm; the highest mark was 56/60. Those of you who scored above 45/60 should be very proud of yourselves.

Some comments about common mistakes and issues from grading the midterm:

• Knowing the definitions remains essential. This includes getting quantifiers in the right order.
• Quantifying things remains essential. If you write "$f(x) \lt 5$ for all $x \gt 0$" it does not follow that "$f(x) \lt 5$". If you write "Let $N$ be so that for all $k \gt N$, $d(x_{n_k}, a) \lt \epsilon$ and $M$ be so that for all $s, m \gt M, d(x_s, x_m) \lt \epsilon$" you can't just write "$d(x_s, a) \leq d(x_s, x_{n_k}) + d(x_{n_k}, a) \lt 2\epsilon$" on the next line; you need to explain what $s$ and $k$ are supposed to be, and why they satisfy the conditions to make the estimates work.

• There is not a different definition of "convergent" for subsequences as opposed to sequences. A sequence $(a_n)_n$ converges to $a$ if for every $\epsilon \gt 0$ there is $N \in \N$ so that for all $n \gt N$, $d(a_n, a) \lt \epsilon$. A subsequence $(x_{k_n})_n$ of a sequence $(x_k)_k$ converges to $a$ if for every $\epsilon \gt 0$ there is $N \in \N$ so that for all $n \gt N$, $d(x_{k_n}, a) \lt \epsilon$, not "for all $k_n \gt N$, $d(x_{k_n}, a) \lt \epsilon$".

  It may be useful to remember that $(a_k)_k$ is a fancy way of writing the function \begin{align*}\N&\longrightarrow M\\k&\longmapsto a_k,\end{align*} so a subsequence $(a_{k_n})_n$ is just a composition of functions: \begin{align*}\N\longrightarrow\N&\longrightarrow M\\n\longmapsto k_n&\longmapsto a_{k_n}.\end{align*} When you want to define a property like convergence, though, you can't peel apart the composition and look inside: you only get to see how each index $n$ maps to a given term $a_{k_n}$. The definition only gets to talk about indices (here, $n$) and terms ($a_{k_n}$).

  A related problem is trying to quantify over an expression rather than a variable. You can say "for all $x$ with $f(x) \gt 0$, blah blah blah"; you can't say "for all $f(x) \gt 0$, blah blah blah". You can say "Let $x \in g^{-1}(U)$ so $g(x) \in U$" or "Let $x \in X$ with $g(x) \in U$", but you shouldn't say "Let $g(x) \in U$" (since, in particular, $U$ may have points not of the form $g(x)$, and it's unclear whether you're trying to ask for a certain $x$, or for $g$ to behave a certain way at a particular point, or both, or something else entirely).

• $(x_n)_n$ cannot stand for an arbitrary subsequence of $(x_k)_k$, it is the sequence $(x_k)_k$.
  • Are the functions $f(x) = x^2$ and $f(t) = t^2$ the same? Yes.
  • Are the numbers $\int_a^b g(t)\,dt$ and $\int_a^b g(x)\,dx$ the same? Yes.
  • Are the sets $\set{a \in \R \mid a \gt a^2}$ and $\set{b \in \R \mid b \gt b^2}$ the same? Yes.
  • Are the sequences $(a_n)_n$ and $(a_k)_k$ the same? Also yes.
  More subtly, $(a_{n_i})_i$ and $(a_{k_i})_i$ are not the same for the same reason that the sequences $(a_i)_i$ and $(b_i)_i$ are not the same (or more poignantly, that $(n_i)_i$ and $(k_i)_i$ are not the same). Also, $(x_n)_{n_i}$ and $(x_{n_i})_{n_i}$ are not subsequences; they are meaningless.
• In terms of general test-taking strategy: I am unlikely to give you unnecessary assumptions. If I say "Suppose $f : [0, \infty) \to \R$ is continuous and $\displaystyle\lim_{t\to\infty}f(t)$ converges", you will probably need to use both of these assumptions to prove what I ask you to prove.
• Several people independently assumed that the function in 4B was monotonically increasing. I'm rather confused about why this was such a popular assumption.