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.