Sequences and subsequences, Part I
Sequences and subsequences play an important role throughout mathematics, and especially in mathematical analysis. In this series (or sequence?) of posts, I will combine some standard theory with some of my own musings on the subject. For the purposes of these posts, note that I work with the convention that $\N=\{1,2,3,\dots\}$, so that $0\notin\N$. Let $X$ be a (non-empty) set. By default I will be working with sequences of the form $(x_n)_{n\in\N}$ where the $x_n$ are elements of $X$. Such a sequence may also be denoted by $(x_n)_{n=1}^\infty$ or $x_1,x_2,x_3,\dots$, or even $(x_n)$ for short (with the default assumption that $n$ runs through $\N$). By abuse of notation we may sometimes write $(x_n)\subseteq X$ for a sequence of elements of $X$, but the sequence $(x_n)_{n\in\N}$ is really not the same thing as the set of values $\{x_n:n\in\N\}.$ We say that two sequences $(x_n)_{n\in\N}$ and $(y_n)_{n\in\N}$ of elements of $X$ are equal if, for all $n\in\N$, we have $x_n=y_n$. ...