$$ \newcommand{\cis}{\operatorname{cis}} \newcommand{\norm}[1]{\left\|#1\right\|} \newcommand{\paren}[1]{\left(#1\right)} \newcommand{\sq}[1]{\left[#1\right]} \newcommand{\abs}[1]{\left\lvert#1\right\rvert} \newcommand{\set}[1]{\left\{#1\right\}} \newcommand{\ang}[1]{\left\langle#1\right\rangle} \newcommand{\floor}[1]{\left\lfloor#1\right\rfloor} \newcommand{\ceil}[1]{\left\lceil#1\right\rceil} \newcommand{\C}{\mathbb{C}} \newcommand{\D}{\mathbb{D}} \newcommand{\R}{\mathbb{R}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\N}{\mathbb{N}} \newcommand{\F}{\mathbb{F}} \newcommand{\T}{\mathbb{T}} \renewcommand{\S}{\mathbb{S}} \newcommand{\intr}{{\large\circ}} \newcommand{\limni}[1][n]{\lim_{#1\to\infty}} \renewcommand{\Re}{\operatorname{Re}} \renewcommand{\Im}{\operatorname{Im}} \newcommand{\cA}{\mathcal{A}} \newcommand{\cB}{\mathcal{B}} \newcommand{\cC}{\mathcal{C}} \newcommand{\cD}{\mathcal{D}} \newcommand{\cE}{\mathcal{E}} \newcommand{\cF}{\mathcal{F}} \newcommand{\cG}{\mathcal{G}} \newcommand{\cH}{\mathcal{H}} \newcommand{\cI}{\mathcal{I}} \newcommand{\cJ}{\mathcal{J}} \newcommand{\cK}{\mathcal{K}} \newcommand{\cL}{\mathcal{L}} \newcommand{\cM}{\mathcal{M}} \newcommand{\cN}{\mathcal{N}} \newcommand{\cO}{\mathcal{O}} \newcommand{\cP}{\mathcal{P}} \newcommand{\cQ}{\mathcal{Q}} \newcommand{\cR}{\mathcal{R}} \newcommand{\cS}{\mathcal{S}} \newcommand{\cT}{\mathcal{T}} \newcommand{\cU}{\mathcal{U}} \newcommand{\cV}{\mathcal{V}} \newcommand{\cW}{\mathcal{W}} \newcommand{\cX}{\mathcal{X}} \newcommand{\cY}{\mathcal{Y}} \newcommand{\cZ}{\mathcal{Z}} \newcommand{\bA}{\mathbb{A}} \newcommand{\bB}{\mathbb{B}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bD}{\mathbb{D}} \newcommand{\bE}{\mathbb{E}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bG}{\mathbb{G}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bI}{\mathbb{I}} \newcommand{\bJ}{\mathbb{J}} \newcommand{\bK}{\mathbb{K}} \newcommand{\bL}{\mathbb{L}} \newcommand{\bM}{\mathbb{M}} \newcommand{\bN}{\mathbb{N}} \newcommand{\bO}{\mathbb{O}} \newcommand{\bP}{\mathbb{P}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bS}{\mathbb{S}} \newcommand{\bT}{\mathbb{T}} \newcommand{\bU}{\mathbb{U}} \newcommand{\bV}{\mathbb{V}} \newcommand{\bW}{\mathbb{W}} \newcommand{\bX}{\mathbb{X}} \newcommand{\bY}{\mathbb{Y}} \newcommand{\bZ}{\mathbb{Z}} $$

Useful links

Office hours:

  • Mondays 18:00-19:00
  • Wednesdays 14:00-16:00

Email

GSI:

Nima Moini
  • Mondays 10:00-14:00
  • Tuesdays 10:00-14:00
  • Wednesdays 10:00-12:00

Exams

Let $(\mathcal{M}, d)$ be a metric space.

  1. Let $(x_n)_n$ be a sequence in $M$. Then $(x_n)_n$ is convergent _____ it is bounded.
    1. if (but not only if)
    2. only if (but not necessarily if)
    3. if and only if
    4. (none of the above is true)
  2. If $(x_n)_n$ is a sequence in $E \subseteq M$ which converges, then $$L := \limni x_n \in \overline{E}.$$ Which of the following is a reasonable way to begin a proof of this fact?
    1. Let $n \in N$, and set $\epsilon = 2d(x_n, L)$.
    2. Let $\epsilon \gt 0$ and choose $N$ large enough that for $n \gt N$, $d(x_n, L) \lt \epsilon$.
    3. Suppose that $L \notin \overline{E}$, and choose $\epsilon \gt 0$ so that $\epsilon \lt d(L, x_n)$.
    4. Let $y \notin \overline{E}$ and let $r = d(y, L) \gt 0$.
  3. If $(x_n)_n$ is a sequence which takes on only finitely many distinct values, it converges.
    1. True.
    2. False.
  4. If $(x_n)_n$ and $(y_n)_n$ are convergent sequences with $$\limni x_n = a = \limni y_n,$$ then $\limni d(x_n, y_n) = 0$. Which of the following inequalitities is true and would be useful in proving this fact?
    1. $d(x_n, a) \leq d(x_n, y_n) + d(y_n, a)$
    2. $d(x_n, y_n) - d(a, a) \leq \min(d(x_n, a), d(y_n, a))$
    3. $d(x_n, y_n) \leq d(x_n, a) + d(a, y_n)$
    4. $d(x_n + y_n, 2a) \leq d(x_n, a) + d(y_n, a)$