Finite Sequences

A sequence is a list of elements in a particular order. While it is natural to picture a sequence of numbers $a_1,a_2,\ldots$ informally, it is more precise — and more useful for proofs — to think of a sequence as a function whose domain is either $\{1,2,\ldots,n\}$ (a finite sequence) or all of $\mathbb N$ (an infinite sequence). Formally: if $X$ is any set and $n\in\mathbb N$, a function $f:\{1,\ldots,n\}\to X$ is a finite sequence on $X$, and a function $g:\mathbb N\to X$ is an infinite sequence on $X$; the value $f(k)$ is written $a_k$, and the sequence itself is denoted $(a_n)$.

If $X$ is a set of real numbers, the sequence is called a numerical sequence (or sequence of real numbers) — the only kind this chapter deals with, referred to simply as "sequences."

• Every sequence is a function, but not every function is a sequence (a sequence's domain must be $\{1,\ldots,n\}$ or $\mathbb N$).
• Unlike a set, whose elements are never repeated, the terms of a sequence may repeat; a sequence all of whose terms are equal is a constant sequence. …