# sequence

## Sequences

Given any set $X$, a 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 $f(0),f(1),f(2)\dots$.

## Generalized sequences

One can generalize the above definition to any arbitrary ordinal. For any set $X$, a generalized sequence or 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 finite sequence.

Title sequence Sequence 2013-03-22 11:50:33 2013-03-22 11:50:33 djao (24) djao (24) 11 djao (24) Definition msc 03E10 msc 40-00 ConvergentSequence generalized sequence transfinite sequence finite sequence