sequenceSequence2001-10-19 13:23:382008-04-22 08:21:21Definitiongeneralized sequencetransfinite sequencefinite sequence\usepackage{amssymb}
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Given any set $X$, a \emph{sequence} in $X$ is a function $f\colon \mathbb{N} \to X$ from the set of natural numbers to $X$. Sequences are usually written with subscript notation: $x_0, x_1, x_2 \dots$, instead of $f(0), f(1), f(2) \dots $.
\paragraph{Generalized sequences}
One can generalize the above definition to any arbitrary ordinal. For any set $X$, a \emph{generalized sequence} or \emph{transfinite sequence} in $X$ is a function $f\colon \omega \to X$ where $\omega$ is any ordinal number. If $\omega$ is a finite ordinal, then we say the sequence is a \emph{finite sequence}.