$$
\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}}
$$
Math 104 Assignment 5 common mistakes
- If $E \subset M$, it may not be the case that all points of $E$ are either limit points or interior points. A point may be both, neither, one, or the other.
For example, if $E = \set{0} \subset \R$ then $0$ is neither a limit point nor an interior point of $E$, and in fact, $E$ has no limit points; on the other hand, if $E = \R$ then $0$ is both an interior point and a limit point of $E$.
- In an arbitrary metric space $(M, d)$, with $x, y \in M$, you can't write $|x-y|$ or $|x, y|$ to mean the distance from $x$ to $y$; write $d(x, y)$.
- The statement "$(x_n)_n$ does not converge to $L$" means "for some $\epsilon \gt 0$, for every $N \in \N$ there is some $n \gt N$ with $d(x_n, L) \geq \epsilon$".
In symbols,
\[\begin{split}&\neg (\forall \epsilon \gt 0 \exists N \in \N \forall n \gt N, d(x_n, L) \lt \epsilon)\\
\Leftrightarrow&\exists\epsilon\gt0\neg(\exists N \in \N \forall n \gt N d(x_n, L) \lt \epsilon)\\
\Leftrightarrow&\exists\epsilon\gt0\forall N \in \N\neg(\forall n \gt N d(x_n, L) \lt \epsilon)\\
\Leftrightarrow&\exists\epsilon\gt0\forall N \in \N\exists n \gt N \neg(d(x_n, L) \lt \epsilon)\\
\Leftrightarrow&\exists\epsilon\gt0\forall N \in \N\exists n \gt N d(x_n, L) \geq \epsilon.\\
\end{split}\]
(I don't know how to typeset symbols and make them look good.)
- Sequences are functions defined on $\N$, and as such, have infinitely many terms.